Research ArticleAnalytical Method of Modelling the Geometric System ofCommunication Route

Wladyslaw Koc

Faculty of Civil and Environmental Engineering Gdansk University of Technology Narutowicza 1112 80-233 Gdansk Poland

Correspondence should be addressed to Wladyslaw Koc kocwlpggdapl

Received 10 October 2013 Accepted 14 April 2014 Published 11 May 2014

Academic Editor Youqing Wang

Copyright copy 2014 Wladyslaw Koc This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The paper presents a new analytical approach to modelling the curvature of a communication route by making use of differentialequations The method makes it possible to identify both linear and nonlinear curvature It enables us to join curves of the sameor opposite signs of curvature Solutions of problems for linear change of curvature and selected variants of nonlinear curvaturein polynomial and trigonometric form were analyzed A comparison of determined horizontal transition curves was made andexamples of negotiating these curves into a geometric system were given

1 Introduction

A rapid progress in computer calculation technique algo-rithms based on current knowledge and inspired by furtherprospecting is also taken into account in the field of geometri-cal shaping of communication routes In the sphere of shapingthe communication routes vehicular roads and railway linesthe development of satellite measuring technique GNSS [1 2]also plays a crucial role A particular significance here will beattached to the question of vehicular dynamicsTherefore themodelling of curvature will obtain a key role

This subject also includes the design of elements of theroute of diversified curvature [3]The elaboration of amethodfor a precise determination of the coordinate route pointsseems to be the most appropriate area of action To performthis activity use is made of tools necessary to provide ana-lytical solutions that is the most advantageous for practicalapplication The junction of elements of a vehicular route orrailway line of diversified curvature should ensure a continu-ous change of the unbalanced side acceleration in an advan-tageous way for interaction dynamics within the road-vehiclesystem This is strictly connected with the proper shaping ofthe curvature

Themost developed investigation branch concerning thisproblem has been for years the study of transition curvesconnecting straight lengths of route with circular arcs Theproblem of transition curves is related to vehicular roads and

railway routes However it is easy to note that there is adistinct disproportion in taking interest in this problem Asearch for new solutions is going on this sphere on vehicularroads (eg [4ndash12]) As regards the railway lines the situationis entirely different Publications on new transition curves arerelatively rare and in the majority were published long ago(eg [13ndash17])

The application of transition curves is aimed at ensuring acontinuous change of the unbalanced side acceleration bet-ween intervals of motor road (or railway route) of diversi-fied curvature in an advantageous way for the interactiondynamicswithin the road-vehicle system Such a requirementrelates to all sorts of transition curves Under this situation itmight appear that there is one particular algorithm to createthem common to all the analyzed curves In fact most of thesolutions that have been known so far appear independentlyand bear various names (sometimes coming from the nameof their author) A general knowledge of the determination oftransition curves equations would make it possible to com-pare various forms of curves with each other and to make anassessment of their practical application

Twenty years ago this issue had already been sufficientlyexplained with respect mainly to railway routes [16] Atten-tion was then concentrated on working out a technique ofidentification of unbalanced accelerations occurring on vari-ous types of transition curves It was based on a comparativeanalysis of some selected transition curves with the use of

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 679817 13 pageshttpdxdoiorg1011552014679817

2 Mathematical Problems in Engineering

a dynamicmodel In themethod accelerationwas a factor thatexcited the transverse vibrations of the carriage [18]Themainconclusion resulting from the considerations was to provethe relation existing between the response of the system andthe class of the excitation functionThe dynamic interactionswere smaller (ie more advantageous) if the class of thefunction was higher It turned out that evidently the largestacceleration values were noted on the cubical parabola (classof function C0) With respect to Bloss curve and a cosinecurve (class C1) they are significantly smaller whereas onsine curve (class C2) they are the smallest

After years the mentioned identification method ofunbalanced accelerations [16] provided a source of inspira-tion for the elaboration of a new technique for modelling thegeometric system of the communication route based on join-ing two points of the route with diversified curvature

2 Analytical Method ofModelling the Curvature

Ameasure of the route bending is the ratio of the angle whichdetermines the direction of the vehiclersquos longitudinal axis aftercovering a certain arc (Figure 1) The curvature of curve 119870 atpoint119872 is called the boundary which is aimed at by the rela-tion of acute angle ΔΘ between tangents to curve119870 at points119872 and119872

1 to the length of arc Δ119897when point119872

1tends along

curve 119870 to point 119872

119896 = limΔ119897rarr0

1003816100381610038161003816100381610038161003816

ΔΘ

Δ119897

1003816100381610038161003816100381610038161003816=

119889Θ

119889119897 (1)

If the operation procedure by the use of rectangular coor-dinates 119909 119910 is to be continued it is necessary to take into con-sideration the sign of the curvature As illustrated in Figure 1angle ΔΘ = Θ

1minus Θ gt 0 the curvature is assumed here that

it has a positive value which as can be seen accompanies thecurves with downward convexities A negative value of cur-vature corresponds to angle ΔΘ lt 0 and appears on curvesof an upward convexity

Making a generalization of the identification methodapplied to unbalanced accelerations occurring on varioustypes of transition curves [16] it is possible to search for cur-vature function 119896(119897) among the solutions of the differentialequation

119896(119898)

(119897) = 119891 [119897 119896 1198961015840 119896

(119898minus1)] (2)

with conditions for the transition curve at the outset (for 119897 =

0) and at the end (for 119897 = 119897119896)

119896(119894)

(0+) =

1198961

for 119894 = 0

0 for 119894 = 1 2 1198991

119896(119895)

(119897minus

119896) =

1198962

for 119895 = 0

0 for 119895 = 1 2 1198992

(3)

where 1198961and 119896

2indicate the curvature magnitudes on both

ends of the curve (taking note of an adequate sign)The orderof the differential equation (2) is 119898 = 119899

1+ 1198992

+ 2 and

Δx

Δy

x

y

M

K

Δl

Θ1Θ

ΔΘ

ΔΘM1

Figure 1 Schematic diagram for the explanation of the notion ofcurvature

the obtained function 119896(119897) is of class C119899 in the range ⟨0 119897119896⟩

where 119899 = min(1198991 1198992)

The application of themethodunder considerationmakesit possible to combine various geometric elements for exam-ple a straightwith a circular arc and also after introducing anadequate sign of curvature two circular arcs of a consistentrun or converse arcs

On determining the curvature function 119896(119897) a fundamen-tal task is to find the coordinates that correspond to a curve inthe rectangular coordinate system 119909 119910 A solution for such asystem is required by the satellite measuring technique GNSS[19ndash21] which becomes an aid in the determination of thecoordinate points in a uniform 3D-system of coordinatesWGS 84 (the world geodetic system 1984) Next themeasuredellipsoidal coordinates (GPS) are transformed to Gauss-Kruger (119883 119884) conformal coordinates [22] An assumption ismade that the beginning of system119909119910 is at the terminal pointof the input curve (with curvature 119896

1) while the axis of

abscissae is tangent to this curve at this pointThe equation of the sought after connection can be

written in parametric form

119909 (119897) = int cosΘ (119897) 119889119897

119910 (119897) = int sinΘ (119897) 119889119897

(4)

Parameter 119897 is the position of a given point along thecurve length FunctionΘ(119897) is determined on the basis of theformula

Θ (119897) = int 119896 (119897) 119889119897 (5)

The presented method of modelling the curvature has auniversal character It can be applied to both vehicular roadsand railway routes In the case of railway routes there arises anadditional possibility for modelling in a similar way thesuperelevation ramp understood here as a determined differ-ence of heights of the railThe principle of operation is similarto curvaturemodellingThe only differences are that 119896

1has to

be replaced by the value of superelevation ℎ1at the beginning

of the superelevation ramp and 1198962has to be replaced by the

value of superelevation ℎ2at the end of the superelevation

Mathematical Problems in Engineering 3

ramp The ramp length of course corresponds to the lengthof the transition curveUsing the sameprocedure it is possibleto find the unbalanced acceleration 119886(119897)

3 Transition Curve of Linear Curvature

31 Finding Out the Curvature Equation Linear change ofcurvature along a defined length 119897

119896is obtained by assuming

two principal conditions

119896 (0+) = 1198961

119896 (119897minus

119896) = 1198962

(6)

and a differential equation

11989610158401015840(119897) = 0 (7)

After the determination of constants the solution of thedifferential problem (6) (7) is as follows

119896 (119897) = 1198961+

1

119897119896

(1198962minus 1198961) 119897 (8)

Function Θ(119897) is obtained from formula (5) In the caseunder consideration

Θ (119897) = 1198961119897 +

1

2119897119896

(1198962minus 1198961) 1198972 (9)

32 The Determination of the Transition Curve Length (forRailway Lines) With regard to rail tracks the transition curvemust satisfy two kinematic conditions

120595max le 120595dop (10)

119891max le 119891dop (11)

where 120595max is the maximum value of acceleration incrementon transition curve inms3120595dop is permissible value of accel-eration increment in ms3 119891max is the maximum speed oflifting the wheel on superelevation ramp in mms and 119891dop ispermissible value of speed of lifting the wheel in mms

The unbalanced acceleration 119886(119897) occurring on the tran-sition curve results from speed V of the train magnitude ofcurvature and ordinates of the superelevation ramp itscourse is similar to 119896(119897) and ℎ(119897) Analogously with (8)

119886 (119897) = 1198861+

1

119897119896

(1198862minus 1198861) 119897 (12)

where 1198861and 119886

2are accelerations at ends of the transition

curve

1198861=

V2

rad1198961minus 119892

ℎ1

119904 119886

2=

V2

rad1198962minus 119892

ℎ2

119904

(13)

(ℎ1and ℎ

2are ordinates of the superelevation ramp at its

ends)On the assumption that the speed of the train V = const

the formula for acceleration increment 120595(119897) is as follows

120595 (119897) = V119889

119889119897119886 (119897) =

V119897119896

(1198862minus 1198861) (14)

As can be seen the acceleration increment is here a con-stant value However from condition (10) it follows that

119897120595

119896ge

V120595dop

10038161003816100381610038161198862 minus 1198861

1003816100381610038161003816 (15)

The superelevation ramp equation in the case under con-sideration is similar to (8) and (12)

ℎ (119897) = ℎ1+

1

119897119896

(ℎ2minus ℎ1) 119897 (16)

The speed of lifting the wheel 119891(119897) on the superelevationramp (assuming that the travelling speed of the train V =const) is determined by the equation

119891 (119897) = V119889

119889119897ℎ (119897) =

V119897119896

(ℎ2minus ℎ1) (17)

Thus also119891(119897) is here a constant value but fromcondition(11) it follows that

119897119891

119896ge

V119891dop

1003816100381610038161003816ℎ2 minus ℎ1

1003816100381610038161003816 (18)

The assumed length of the transition curvemust fulfill thecondition

119897119896ge max (119897

120595

119896 119897119891

119896) (19)

33 Coordinates of Transition Curve Connecting Uniform Cur-vatures The determination of 119909(119897) and 119910(119897) by using (4) willneed the expansion of integrands intoMaclaurin series Aftercompleting the whole procedure the following parametricequations are obtained

119909 (119897) = int cosΘ (119897) 119889119897

= 119897 minus1198962

1

61198973minus

1198961

8119897119896

(1198962minus 1198961) 1198974

+ [1198964

1

120minus

1

401198972

119896

(1198962minus 1198961)2] 1198975+

1198963

1

72119897119896

(1198962minus 1198961) 1198976

minus [1198966

1

5040minus

1198962

1

1121198972

119896

(1198962minus 1198961)2] 1198977

minus [1198965

1

1920119897119896

(1198962minus 1198961) minus

1198961

3841198973

119896

(1198962minus 1198961)3] 1198978

+ [1198968

1

362880minus

1198964

1

17281198972

119896

(1198962minus 1198961)2

+1

34561198974

119896

(1198962minus 1198961)4] 1198979sdot sdot sdot

4 Mathematical Problems in Engineering

119910 (119897) = int sinΘ (119897) 119889119897

=1198961

21198972+

1

6119897119896

(1198962minus 1198961) 1198973minus

1198963

1

241198974minus

1198962

1

20119897119896

(1198962minus 1198961) 1198975

+ [1198965

1

720minus

1198961

481198972

119896

(1198962minus 1198961)2] 1198976

+ [1198964

1

336119897119896

(1198962minus 1198961) minus

1

3361198973

119896

(1198962minus 1198961)3] 1198977

minus [1198967

1

40320minus

1198963

1

3841198972

119896

(1198962minus 1198961)2] 1198978

minus [1198966

1

12960119897119896

(1198962minus 1198961) minus

1198962

1

8641198973

119896

(1198962minus 1198961)3] 1198979+ sdot sdot sdot

(20)

Equations (20) for 1198961= 0 describe the curve in the form

of clothoid used to connect a straight with a circular arc

34 Coordinates of Transition Curve Connecting Inverse Cur-vatures At inverse arcs (ie with diversified curvature signs)there arises the problem related to the determined coordi-nates 119909(119897) and 119910(119897) due to the course of function Θ(119897) Forsign 119896

1= sign 119896

2function Θ(119897) is a monotonic function

whereas for sign 1198961

= sign 1198962in the diagram of function Θ(119897)

there appears an extremum at point 1198970 where

Θ1015840(1198970) = 119896 (119897

0) = 1198961+ (1198962minus 1198961)1198970

119897119896

= 0 (21)

Values 1198970and Θ(119897

0) follow from the relations

1198970= minus

1198961

1198962minus 1198961

119897119896

Θ(1198970) = minus

1198962

1

2 (1198962minus 1198961)119897119896

(22)

Figure 2 illustrates a scheme of angle Θ(119897) for the con-nection of inverse circular arcs of radii 119877

1= 1200m and

1198772= 700mThe length of the transition curve 119897

119896= 60m was

determined for speed value V = 90 kmh using the procedureas described at Section 32 (the assumed superelevationvalues were ℎ

1= 20mm and ℎ

2= 45mm)

In this situation the parametric equations of transitioncurve (20) are effective for 119897 isin ⟨0 119897

0⟩ for 119897 isin (119897

0 119897119896⟩ functions

0

0005

001

0015

0 20 40 60 80minus0005

minus001

minus0015

minus002

Q(rad)

l (m)

Figure 2 Diagram of angle Θ(119897) defined by (9) for circulararcs connection of curvatures 119896

1= 11200 radm and 119896

2=

minus1700 radm by means of transition curve of length 119897119896= 60m

cosΘ(119897) and sinΘ(119897) should be expanded to Taylor seriesAfter integration of the equations take the following form

119909 (119897) = int cosΘ (119897) 119889119897

= 119909 (1198970) + cos(minus

1

2

1198962

1

1198962minus 1198961

119897119896) (119897 minus 119897

0)

minus [1

6119897119896

(1198962minus 1198961) sin(minus

1

2

1198962

1

1198962minus 1198961

119897119896)] (119897 minus 119897

0)3

minus [1

401198972

119896

(1198962minus 1198961)2 cos(minus

1

2

1198962

1

1198962minus 1198961

119897119896)] (119897 minus 119897

0)5

+ [1

3361198973

119896

(1198962minus 1198961)3 sin(minus

1

2

1198962

1

1198962minus 1198961

119897119896)] (119897 minus 119897

0)7

+ sdot sdot sdot

(23)

119910 (119897) = int sinΘ (119897) 119889119897

= 119910 (1198970) + sin(minus

1

2

1198962

1

1198962minus 1198961

119897119896) (119897 minus 119897

0)

+ [1

6119897119896

(1198962minus 1198961) cos(minus

1

2

1198962

1

1198962minus 1198961

119897119896)] (119897 minus 119897

0)3

minus [1

401198972

119896

(1198962minus 1198961)2 sin(minus

1

2

1198962

1

1198962minus 1198961

119897119896)] (119897 minus 119897

0)5

minus [1

3361198973

119896

(1198962minus 1198961)3 cos(minus

1

2

1198962

1

1198962minus 1198961

119897119896)] (119897 minus 119897

0)7

+ [1

34561198974

119896

(1198962minus 1198961)4 sin(minus

1

2

1198962

1

1198962minus 1198961

119897119896)] (119897 minus 119897

0)9

+ sdot sdot sdot

(24)

Mathematical Problems in Engineering 5

On completing the parametric equations (20) and (23)along with (24) a correct solution of the task for the caseof diversified curvature signs is obtained (Figure 3) From apractical point of view the determination of the magnitude ofthe tangent slope angle at the end of the transition curve isvery important It amounts to

Θ(119897119896) =

1

2(1198961+ 1198962) 119897119896 (25)

A knowledge of Θ(119897119896) makes it possible to add the tran-

sition curve to the other one having curvature 1198962at its initial

point (satisfying the tangency condition of both the curves)

4 Transition Curve of Polynomial Curvature

From the point of view of vehiclesrsquo dynamics the transitioncurve of linear change of curvature is not the most advan-tageous solution [23] A definitely better one is nonlinearsolution whose characteristics are interesting enough to becomparedwith the Bezier curves preferable lately in literature[4 7]

The differential equation (2) allows to appoint an unlim-ited number of curves with nonlinear curvature change Inthe case of geometric layout of communication routes it willbe appropriate to consider the transition curves of poly-nomial and trigonometric curvature In fact such forms ofcurves are used in the standard ways to connect a straight linewith a circular arc

41 Formulation of the Curvature Equation To formulate thecurvature equation in the polynomial form let us assume thefollowing boundary conditions

119896 (0+) = 1198961 119896 (119897

minus

119896) = 1198962

1198961015840(0+) = 0 119896

1015840(119897minus

119896) = 0

(26)

and the differential equation

119896(4)

(119897) = 0 (27)

After solving the differential problem (26) (27) thefollowing formula for curvature 119896(119897) is obtained

119896 (119897) = 1198961+

3

1198972

119896

(1198962minus 1198961) 1198972minus

2

1198973

119896

(1198962minus 1198961) 1198973 (28)

The form of function Θ(119897) determined on the basis of (5)and (28) is as follows

Θ (119897) = 1198961119897 +

1

1198972

119896

(1198962minus 1198961) 1198973minus

1

21198973

119896

(1198962minus 1198961) 1198974 (29)

On rail routes the length of the transition curve isdetermined like at Section 32 for solution of linear curvatureThe following conditions should be fulfilled here

119897120595

119896ge

3

2

V120595dop

10038161003816100381610038161198862 minus 1198861

1003816100381610038161003816

119897119891

119896ge

3

2

V119891dop

1003816100381610038161003816ℎ2 minus ℎ1

1003816100381610038161003816

(30)

0

01

02

03

04

0 20 40 60 80x (m)

y(m

)

Figure 3 Diagram of horizontal ordinates 119910(119909) defined by (20)(23) and (24) for the joint of circular arcs of curvatures 119896

1=

11200 radm and 1198962

= minus1700 radm with transition curve oflength 119897

119896= 60m (using different horizontal and vertical scales)

The adopted length of the transition curve must satisfycondition (19)

42 Coordinates of Transition Curve Connecting Uniform Cur-vatures On expanding functions cosΘ(119897) and sinΘ(119897) intoMaclaurin series and after integration the following paramet-ric equations are acquired

119909 (119897) = int cosΘ (119897) 119889119897

= 119897 minus1198962

1

61198973+ [

1198964

1

120minus

1198961

51198972

119896

(1198962minus 1198961)] 1198975

+ [1198961

121198973

119896

(1198962minus 1198961)] 1198976

minus [1198966

1

5040minus

1198963

1

421198972

119896

(1198962minus 1198961) +

1

141198974

119896

(1198962minus 1198961)2] 1198977

minus [1198963

1

961198973

119896

(1198962minus 1198961) minus

1

161198975

119896

(1198962minus 1198961)2] 1198978

+ [1198968

1

362880minus

1198965

1

10801198972

119896

(1198962minus 1198961) +

1198962

1

361198974

119896

(1198962minus 1198961)2

+1

721198976

119896

(1198962minus 1198961)2] 1198979+ sdot sdot sdot

119910 (119897) = int sinΘ (119897) 119889119897

=1198961

21198972minus [

1198963

1

24minus

1

41198972

119896

(1198962minus 1198961)] 1198974

minus1

101198973

119896

(1198962minus 1198961) 1198975+ [

1198965

1

720minus

1198962

1

121198972

119896

(1198962minus 1198961)] 1198976

+1198962

1

281198973

119896

(1198962minus 1198961) 1198977

6 Mathematical Problems in Engineering

0

00005

0001

0 20 40 60 80 100l (m)minus00005

minus0001

minus00015

minus0002

k(radm

)

Figure 4 Diagram of curvature 119896(119897) described by (28) for theconnection of circular arcs of curvatures 119896

1= 11200 radm and

1198962

= minus1700 radm with the use of transition curve of length 119897119896

=

90m

minus [1198967

1

40320minus

1198964

1

1921198972

119896

(1198962minus 1198961) +

1198961

161198974

119896

(1198962minus 1198961)2] 1198978

+ [1198961

181198975

119896

(1198962minus 1198961)2minus

1198964

1

4321198973

119896

(1198962minus 1198961)] 1198979+ sdot sdot sdot

(31)

Equations (31) for 1198961

= 0 describe the so-called Blosscurve which can be used to join a straight to a circular arc

43 Coordinates of Transition Curve Connecting Inverse Cur-vatures When the arcs are inverse (ie the curvature signsare different Figure 4) there occurs a problem relating to thedetermined coordinates 119909(119897) and 119910(119897) which results fromfunction Θ(119897) For sign 119896

1= sign 119896

2function Θ(119897) is a mono-

tonic function while for sign 1198961

= sign 1198962in the diagram of

function Θ(119897) there appears extremum at point 1198970 where

Θ1015840(1198970) = 119896 (119897

0) = 1198961+

3

1198972

119896

(1198962minus 1198961) 1198972

0minus

2

1198973

119896

(1198962minus 1198961) 1198973

0= 0

(32)

Parametric equations of transition curve (31) are valid for119897 isin ⟨0 119897

0⟩The determination of value 119897

0makes it necessary to

solve (32) which is cubic The sought-after value 1198970

isin ⟨0 119897119896⟩

satisfying conditions of the problem is defined by the formula

1198970= [

1

2minus cos(

120593

3+

120587

3)] 119897119896 (33)

where angle 120593 follows from the relation

cos120593 =1198961+ 1198962

1198962minus 1198961

(34)

Under such circumstances in order to determine para-metric equations of the transition curve for 119897 isin (119897

0 119897119896⟩ func-

tions cosΘ(119897) and sinΘ(119897) should be expanded into Taylor

series after integration the following parametric equations ofthe transition curve are obtained

119909 (119897) = int cosΘ (119897) 119889119897

= 119909 (1198970) + cosΘ(119897

0) (119897 minus 119897

0)

minus1

6sinΘ(119897

0)Θ10158401015840(1198970) (119897 minus 119897

0)3

minus1

24sinΘ(119897

0)Θ101584010158401015840

(1198970) (119897 minus 119897

0)4

minus 1

40cosΘ(119897

0) [Θ10158401015840(1198970)]2

+1

120sinΘ(119897

0)Θ(4)

(1198970) (119897 minus 119897

0)5

minus1

72cosΘ(119897

0)Θ10158401015840(1198970)Θ101584010158401015840

(1198970) (119897 minus 119897

0)6

+ 1

336sinΘ(119897

0) [Θ10158401015840(1198970)]3

minus1

336cosΘ(119897

0)Θ10158401015840(1198970)Θ(4)

(1198970)

minus1

504cosΘ(119897

0) [Θ101584010158401015840

(1198970)]2

(119897 minus 1198970)7

+ 1

384sinΘ(119897

0) [Θ10158401015840(1198970)]2

Θ101584010158401015840

(1198970)

minus1

1152cosΘ(119897

0)Θ101584010158401015840

(1198970)Θ(4)

(1198970) (119897 minus 119897

0)8

+ 1

3456cosΘ(119897

0) [Θ10158401015840(1198970)]4

+1

1296sinΘ(119897

0)Θ10158401015840(1198970) [Θ101584010158401015840

(1198970)]2

+1

1728sinΘ(119897

0) [Θ10158401015840(1198970)]2

Θ(4)

(1198970)

minus1

10368cosΘ(119897

0) [Θ(4)

(1198970)]2

(119897 minus 1198970)9+ sdot sdot sdot

119910 (119897) = int sinΘ (119897) 119889119897

= 119910 (1198970) + sinΘ(119897

0) (119897 minus 119897

0)

+1

6[cosΘ(119897

0)Θ10158401015840(1198970)] (119897 minus 119897

0)3

+1

24[cosΘ(119897

0)Θ101584010158401015840

(1198970)] (119897 minus 119897

0)4

minus 1

40sinΘ(119897

0) [Θ10158401015840(1198970)]2

minus1

120cosΘ(119897

0)Θ(4)

(1198970) (119897 minus 119897

0)5

minus1

72[sinΘ(119897

0)Θ10158401015840(1198970)Θ101584010158401015840

(1198970)] (119897 minus 119897

0)6

Mathematical Problems in Engineering 7

minus 1

336cosΘ(119897

0) [Θ10158401015840(1198970)]3

+1

336sinΘ(119897

0)Θ10158401015840(1198970)Θ(4)

(1198970)

+1

504sinΘ(119897

0) [Θ101584010158401015840

(1198970)]2

(119897 minus 1198970)7

minus 1

384cosΘ(119897

0) [Θ10158401015840(1198970)]2

Θ101584010158401015840

(1198970)

+1

1152sinΘ(119897

0)Θ101584010158401015840

(1198970)Θ(4)

(1198970) (119897 minus 119897

0)8

+ 1

3456sinΘ(119897

0) [Θ10158401015840(1198970)]4

minus1

1296cosΘ(119897

0)Θ10158401015840(1198970) [Θ101584010158401015840

(1198970)]2

minus1

1728cosΘ(119897

0) [Θ10158401015840(1198970)]2

Θ(4)

(1198970)

minus1

10368sinΘ(119897

0) [Θ(4)

(1198970)]2

(119897 minus 1198970)9+ sdot sdot sdot

(35)

where

Θ(1198970) = 11989611198970+

1

1198972

119896

(1198962minus 1198961) 1198973

0minus

1

21198973

119896

(1198962minus 1198961) 1198974

0

Θ10158401015840(1198970) =

6

1198972

119896

(1198962minus 1198961) 1198970minus

6

1198973

119896

(1198962minus 1198961) 1198972

0

Θ101584010158401015840

(1198970) =

6

1198972

119896

(1198962minus 1198961) minus

12

1198973

119896

(1198962minus 1198961) 1198970

Θ(4)

(1198970) = minus

12

1198973

119896

(1198962minus 1198961)

(36)

Making use of parametric equations (31) and also (35) onecan obtain a correct solution of the problem with diversifiedsigns of curvature

Value of the tangent slope at the end of the transitioncurvemakes it possible to connect to it a curve of curvature 119896

2

satisfying simultaneously the tangency condition of both thecurves The above value is the same as in the case of linearcurvature It is defined by (25)

5 Transition Curve of Curvature inTrigonometric Form

51 Formulation of Curvature Equation As before boundaryconditions (26) and another differential equation areassumed

119896(4)

(119897) +1205872

1198972

119896

11989610158401015840(119897) = 0 (37)

As soon as the differential problem (26) (37) has beensolved a formula for curvature is obtained

119896 (119897) =1

2(1198961+ 1198962) minus

1

2(1198962minus 1198961) cos 120587

119897119896

119897 (38)

The form of function Θ(119897) determined by the use of (5)and (38) is as follows

Θ (119897) =1

2(1198961+ 1198962) 119897 minus

1

2(1198962minus 1198961)119897119896

120587sin 120587

119897119896

119897 (39)

The length of the transition curve on railway routesfollows from conditions

119897120595

119896ge

120587

2

V120595dop

10038161003816100381610038161198862 minus 1198861

1003816100381610038161003816

119897119891

119896ge

120587

2

V119891dop

1003816100381610038161003816ℎ2 minus ℎ1

1003816100381610038161003816

(40)

The assumed length of the curve must satisfy the condi-tion (19)

52 Ordinates of Transition Curve Connecting UniformCurva-tures After the expansion of functions cosΘ(119897) and sinΘ(119897)

intoMaclaurin series to be next integrated the following para-metric equations are obtained

119909 (119897) = int cosΘ (119897) 119889119897

= 119897 minus1198962

1

61198973+ [

1198964

1

120minus

1

60

1205872

1198972

119896

1198961(1198962minus 1198961)] 1198975

minus [1198966

1

5040minus

1

504

1205872

1198972

119896

1198963

1(1198962minus 1198961)

minus1

1680

1205874

1198974

119896

1198961(1198962minus 1198961) minus

1

2016

1205874

1198974

119896

(1198962minus 1198961)2] 1198977

+ [1198968

1

362880minus

1

12960

1205872

1198972

119896

1198965

1(1198962minus 1198961)

minus1

5184

1205874

1198974

119896

1198962

1(1198962minus 1198961)2

minus1

12960

1205874

1198974

119896

1198963

1(1198962minus 1198961) minus

1

90720

1205876

1198976

119896

1198961(1198962minus 1198961)

+1

25920

1205876

1198976

119896

(1198962minus 1198961)2] 1198979+ sdot sdot sdot

8 Mathematical Problems in Engineering

119910 (119897) = int sinΘ (119897) 119889119897

=1

211989611198972minus [

1

241198963

1minus

1

48

1205872

1198972

119896

(1198962minus 1198961)] 1198974

+ [1

7201198965

1minus

1

1441198962

1

1205872

1198972

119896

(1198962minus 1198961)

minus1

1440

1205874

1198974

119896

(1198962minus 1198961)] 1198976

minus [1

403201198967

1minus

1

23041198964

1

1205872

1198972

119896

(1198962minus 1198961)

+1

2304

1205874

1198974

119896

1198961(1198962minus 1198961)2minus

1

3840

1205874

1198974

119896

1198962

1(1198962minus 1198961)

+1

80640

1205876

1198976

119896

(1198962minus 1198961)] 1198978+ sdot sdot sdot

(41)

Equations (41) for 1198961

= 0 describe the curve known ascosinusoid which can be used to connect a straight with acircular arc

53 Transition Curve Coordinates Connecting Inverse Curva-tures Near inverse arcs in the diagram of functionΘ(119897) thereappears the extremum at point 119897

0 where

Θ1015840(1198970) = 119896 (119897

0) =

1

2(1198961+ 1198962) minus

1

2(1198962minus 1198961) cos 120587

119897119896

1198970= 0

(42)

The transition curve parametric equations (41) are effec-tive for 119897 isin ⟨0 119897

0⟩ To determine value 119897

0one must solve (42)

The equation can be written in the form

cos 120587

119897119896

1198970=

1198961+ 1198962

1198962minus 1198961

(43)

It is easy to prove that for sign 1198961

= sign 1198962it is necessary

to satisfy the condition |1198961+1198962| lt |1198962minus1198961| thus |(119896

1+1198962)(1198962minus

1198961)| lt 1 The formula for length 119897

0is as follows

1198970=

1

120587(arccos1198961 + 119896

2

1198962minus 1198961

) 119897119896 (44)

For 119897 isin (1198970 119897119896⟩ functions cosΘ(119897) and sinΘ(119897) should

be expanded into Taylor series The integration provides thefollowing equations

119909 (119897) = int cosΘ (119897) 119889119897

= 119909 (1198970) + cosΘ(119897

0) (119897 minus 119897

0)

minus1

6sinΘ(119897

0)Θ10158401015840(1198970) (119897 minus 119897

0)3

minus1

24sinΘ(119897

0)Θ101584010158401015840

(1198970) sdot (119897 minus 119897

0)4

minus 1

40cosΘ(119897

0) [Θ10158401015840(1198970)]2

+1

120sinΘ(119897

0)Θ(4)

(1198970) (119897 minus 119897

0)5

minus 1

72cosΘ(119897

0)Θ10158401015840(1198970)Θ101584010158401015840

(1198970)

+1

720sinΘ(119897

0)Θ(5)

(1198970) (119897 minus 119897

0)6

+ 1

336sinΘ(119897

0) [Θ10158401015840(1198970)]3

minus1

336cosΘ(119897

0)Θ10158401015840(1198970)Θ(4)

(1198970)

minus1

504cosΘ(119897

0) [Θ101584010158401015840

(1198970)]2

minus1

5040sinΘ(119897

0)Θ(6)

(1198970) (119897 minus 119897

0)7

+ 1

384sinΘ(119897

0) [Θ10158401015840(1198970)]2

Θ101584010158401015840

(1198970)

minus1

1920cosΘ(119897

0)Θ10158401015840(119897) Θ(5)

(119897)

minus1

1152cosΘ(119897

0)Θ101584010158401015840

(1198970)Θ(4)

(1198970)

minus1

40320sinΘ(119897

0)Θ(7)

(1198970) (119897 minus 119897

0)8

+ 1

3456cosΘ(119897

0) [Θ10158401015840(1198970)]4

+1

1296sinΘ(119897

0)Θ10158401015840(1198970) [Θ101584010158401015840

(1198970)]2

+1

1728sinΘ(119897

0) [Θ10158401015840(1198970)]2

Θ(4)

(1198970)

minus1

12960cosΘ(119897

0)Θ10158401015840(1198970)Θ(6)

(1198970)

minus1

6480cosΘ(119897

0)Θ101584010158401015840

(1198970)Θ(5)

(1198970)

minus1

10368cosΘ(119897

0) [Θ(4)

(1198970)]2

minus1

362880sinΘ(119897

0)Θ(8)

(1198970) (119897 minus 119897

0)9+ sdot sdot sdot

Mathematical Problems in Engineering 9

119910 (119897) = int sinΘ (119897) 119889119897

= 119910 (1198970) + sinΘ(119897

0) (119897 minus 119897

0)

+1

6cosΘ(119897

0)Θ10158401015840(1198970) (119897 minus 119897

0)3

+1

24cosΘ(119897

0)Θ101584010158401015840

(1198970) sdot (119897 minus 119897

0)4

minus 1

40sinΘ(119897

0) [Θ10158401015840(1198970)]2

minus1

120cosΘ(119897

0)Θ(4)

(1198970) (119897 minus 119897

0)5

minus 1

72sinΘ(119897

0) 11987610158401015840(1198970)Θ101584010158401015840

(1198970)

minus1

720cosΘ(119897

0)Θ(5)

(1198970) (119897 minus 119897

0)6

minus 1

336cosΘ(119897

0) [Θ10158401015840(1198970)]3

+1

336sinΘ(119897

0)Θ10158401015840(1198970)Θ(4)

(1198970)

+1

504sinΘ(119897

0) [Θ101584010158401015840

(1198970)]2

minus1

5040cosΘ(119897

0)Θ(6)

(1198970) (119897 minus 119897

0)7

minus 1

384cosΘ(119897

0) [Θ10158401015840(1198970)]2

Θ101584010158401015840

(1198970)

+1

1920sinΘ(119897

0)Θ10158401015840(1198970)Θ(5)

(1198970)

+1

1152sinΘ(119897

0)Θ101584010158401015840

(1198970)Θ(4)

(1198970)

minus1

40320cosΘ(119897

0)Θ(7)

(1198970) (119897 minus 119897

0)8

+ 1

3456sinΘ(119897

0) [Θ10158401015840(1198970)]4

minus1

1296cosΘ(119897

0)Θ10158401015840(1198970) [Θ101584010158401015840

(1198970)]2

minus1

1728cosΘ(119897

0) [Θ10158401015840(1198970)]2

Θ(4)

(1198970)

minus1

12960sin119876 (119897

0) 11987610158401015840(1198970) 119876(6)

(1198970)

minus1

6480sin119876 (119897

0) 119876101584010158401015840

(1198970) 119876(5)

(1198970)

minus1

10368sinΘ(119897

0) [Θ(4)

(1198970)]2

+1

362880cosΘ(119897

0)Θ(8)

(1198970) (119897 minus 119897

0)9+ sdot sdot sdot

(45)

where

Θ(1198970) =

1

2(1198961+ 1198962) 1198970plusmn

119897119896

120587radicminus11989611198962

Θ10158401015840(1198970) = ∓

120587

119897119896

radicminus11989611198962

Θ10158401015840(1198970) = ∓

120587

119897119896

radicminus11989611198962

Θ101584010158401015840

(1198970) =

1

2

1205872

1198972

119896

(1198961+ 1198962)

Θ(4)

(1198970) = plusmn

1205873

1198973

119896

radicminus11989611198962

Θ(5)

(1198970) = minus

1

2

1205874

1198974

119896

(1198961+ 1198962)

Θ(6)

(1198970) = ∓

1205875

1198975

119896

radicminus11989611198962

Θ(7)

(1198970) =

1

2

1205876

1198976

119896

(1198961+ 1198962)

Θ(8)

(1198970) = plusmn

1205877

1198977

119896

radicminus11989611198962

(46)

In expressions with double signs plusmn and ∓ the upper onecorresponds to the case 119896

1gt 0 119896

2lt 0 while the lower one

refers to 1198961lt 0 119896

2gt 0

The value of tangent slope at the end of the transitioncurvemakes it possible to connect to it a curve of curvature 119896

2

satisfying the tangency condition of both the curves Thevalue is the same as in the previous cases It is determinedby (25)

6 Comparison of Horizontal Ordinates ofDetermined Transition Curves

61 The Shaping of Horizontal Ordinates of ComparableCurves If the analysis is carried out for railway routes thenthe necessity of meeting kinematic conditions (10) and (11)will require a diversification of the lengths of the comparedwith each other curves (using the same speed of trains)Moreover it is necessary to individually consider cases ofconnecting homogeneous curvatures (sign 119896

1= sign 119896

2) and

inverse arcs (sign 1198961

= sign 1198962)

It is assumed that at joints of consistent arcs with radii1198771= 1200m and 119877

2= 700m an attempt is made to attain a

speed V = 110 kmh Hence it is necessary to employ on arcssuperelevation of ℎ

1= 80mmand ℎ

2= 115mm respectively

The retention of the same kinematic parameters can providediversified lengths of the transition curve

(i) for a curve of linear curvature 119897119896= 40m

(ii) for a curve of polynomial curvature 119897119896= 60m

(iii) for a curve of trigonometric curvature 119897119896= 62 832m

10 Mathematical Problems in Engineering

The above length relations result directly from the equa-tions given at Sections 32 41 and 51

Figure 5 gives diagrams of horizontal ordinates for com-pared transition curves of positive and negative curvaturevalues As seen the curvature ordinates of linear curvatureevidently deviate from the ordinates of other curves In turncurves of polynomial and trigonometric curvature are inprinciple consistent A difference can only be noted in thefinal area And so the class of the function describing thecurvature plays here a significant part with respect to a curveof linear curvature it is a function of class C0 while regardingthe other two curves it is a function of class C1

At the joint of inverse arcs of radii 1198771= 1200m and 119877

2=

700m an attempt is made to reach the speed of V = 90 kmhHence it follows that it is necessary to apply superelevationalong arcs attaining ℎ

1= 20mm and ℎ

2= 45mm respec-

tively By the use the same kinematic parameters it is possibleto obtain the following lengths of the transition curve

(i) for a curve of linear curvature 119897119896= 60m

(ii) for a curve of polynomial curvature 119897119896= 90m

(iii) for a curve of trigonometric curvature 119897119896= 94 248m

Figure 6 presents diagrams of horizontal ordinates ofcompared transition curves for positive and negative valuesof curvature

As can be seen like in the case of consistent arcs the curveordinates of linear curvature deviate significantly from theordinates of the remaining curves but the shape of thesecurves also differs one from the other The class of functiondescribing the curvature is still important However in therange of the same class various geometric solutions arepossible

62 Negotiation of Transition Curves in a Geometric SystemThe transition curves do not appear alone in a geometricsystem but must be connected with some arcs in the neigh-borhood If the input arc is circular then its equation is asfollows

(i) for 1198961gt 0

119910 (119909) = 1198771minus radic1198772

1minus 1199092 (47)

(ii) for 1198961lt 0

119910 (119909) = minus (1198771minus radic1198772

1minus 1199092) (48)

where 119909 lt 0

However if the output arc is also a circular arc then it isexpressed in the following equation

(i) for 1198962gt 0

119910 (119909) = 119910119878minus radic1198772

2minus (119909119878minus 119909)2

119909119878= 119909119870

minus119904119870

radic1 + 1199042

119870

1198772 119910

119878= 119910119870

+1

radic1 + 1199042

119870

1198772

(49)

0051

152

25

0 10 20 30 40 50 60 70x (m)

y(m

)

minus05

minus1

minus15

minus2

minus25

Figure 5 Diagrams of horizontal ordinates of curves joiningconsistent arcs for positive and negative values of curvature (usingdifferent horizontal and vertical scales) curve of linear curvatureline in violet colour curve of polynomial curvature in green colourand curve of trigonometric curvature in red colour

0

05

1

15

0 20 40 60 80 100x (m)

y(m

)

minus05

minus1

minus15

Figure 6 Diagrams of horizontal ordinates of curves connectinginverse arcs for positive and negative values of curvature (usingdifferent horizontal and vertical scales) the curve of linear curvatureis in violet color the curve of polynomial curvature is in green colorand the curve of trigonometric curvature is in red color

(ii) for 1198962lt 0

119910 (119909) = 119910119878+ radic1198772

2minus (119909119878minus 119909)2

119909119878= 119909119870

+119904119870

radic1 + 1199042

119870

1198772 119910

119878= 119910119870

minus1

radic1 + 1199042

119870

1198772

(50)

where 119909 gt 119909119870

In (49) and (50) 119909119870denotes the abscissae of the transition

curve end119910119870is the ordinate of the transition curve endwhile

119904119870is the value of tangent at the end of the transition curve

Values for numerical examples under consideration are givenin Table 1

Figures 7 and 8 illustrate the geometric systems made upof two circular arcs of consistent curvature connected withvarious types of transition curves As seen the application of agiven type of transition curve leads to an adequate position ofthe output circular arcThe position of this arc by the use of alinear transition curve clearly deviates from such cases whereadvantage is taken of curves with polynomial and trigono-metrical curvatureWith regard to the latter amutual positionof the circular arc varies insignificantly

Mathematical Problems in Engineering 11

Table 1 Comparison of values 119909119870 119910119870 and 119904

119870for numerical examples analyzed

Curve 119897119896[m] Θ(119897

119896) [rad] 119904

119870119909119870[m] 119910

119870[m]

Case 1 1198961= 11200 radm and 119896

2= 1700 radm

Curve C0 40 0045238 0045269 39988 0825Curve C1 (p) 60 0067857 0067961 59963 1821Curve C1 (t) 62832 0071060 0071180 62796 1940

Case 2 1198961= 11200 radm and 119896

2= minus1700 radm

Curve C0 60 minus0017857 minus0017859 59999 0143Curve C1 (p) 90 minus0026786 minus0026792 89991 0627Curve C1 (t) 94248 minus0028050 minus0028057 94238 0714

Case 3 1198961= minus11200 radm and 119896

2= 1700 radm

Curve C0 60 001786 0017859 59999 minus0143Curve C1 (p) 90 0026786 0026792 89991 minus0627Curve C1 (t) 94248 0028050 0028057 94238 minus0714

Case 4 1198961= minus11200 radm and 119896

2= minus1700 radm

Curve C0 40 minus0045238 minus0045269 39988 minus0825Curve C1 (p) 60 minus0067857 minus0067961 59969 minus1821Curve C1 (t) 62832 minus0071060 minus0071180 62976 minus1940

0

1

2

3

4

5

6

0 20 40 60 80 100 120x (m)

y(m

)

minus40 minus20

Figure 7 Examples of geometric systems consisting of two circulararcs with consistent curvature (brown line) connected with differenttransition curves for positive values of curvature (using differenthorizontal and vertical scales) curve of linear curvature in violetline curve of polynomial curvature in green and curve of trigono-metric curvature in red

00 20 40 60 80 100 120

x (m)

y(m

)

minus40 minus20

minus1

minus2

minus3

minus4

minus5

minus6

minus7

Figure 8 Examples of geometric systems consisting of two circulararcs with consistent curvature (brown line) connected with differenttransition curves for negative values of curvature (using differenthorizontal and vertical scales) curve of linear curvature in violetline curve of polynomial curvature in green and curve of trigono-metric curvature in red

0

05

1

15

0 50 100 150x (m)

y(m

)

minus05

minus1

minus15

minus2

minus50

Figure 9 Examples of geometric systems consisting of two inversecircular arcs (brown line) connected with different types of transi-tion curves for positive values of input curvature (using differenthorizontal and vertical scales) curve of linear curvature in violetline curve of polynomial curvature in green and curve of trigono-metric curvature in red

Figures 9 and 10 illustrate geometric systems made up oftwo inverse circular arcs connected with various types oftransition curves

Here the layout of the output circular arc is much morediversifiedThe position of the arc obtained by the applicationof the transition curve of linear curvature diverts by severaldozenmeters from situationswhere curves of polynomial andtrigonometric curvature have been used The difference forcurves of polynomial and trigonometric curvature attains inthe numerical example analyzed an order of several meters

The presented examples are taken in the very nature ofthings at random They reveal wide possibilities for the pre-sented analytical method in modelling geometric systems Itsapplication makes it possible to generate diversified layout ofthe communication routes adapted to the field conditions andother circumstances Simultaneously the method ensurescorrectness of the obtained solution in view of the kinematicparameters magnitudes in force

12 Mathematical Problems in Engineering

0

05

1

15

2

0 50 100 150x (m)

y(m

)

minus05

minus1

minus15

minus50

Figure 10 Examples of geometric systems consisting of twoinverse circular arcs (brown line) connected with different typesof transition curves for negative values of input curvature (usingdifferent horizontal and vertical scales) curve of linear curvaturein violet line curve of polynomial curvature in green and curve oftrigonometric curvature in red

7 Conclusions

The presented universal method of modelling geometricalsystem of a communication route is based on the determina-tion of an adequate curvature equation by using differentialequationsThe technique enables us to design connections ofvarious geometrical elements of the route indicating diversifi-cation in its curvature It is characterized by an analytical wayof recording certain functions and moreover it is possible tospecify in advance the class of these functions

The sought-after connection is recorded in the form ofparametric equations 119909(119897) and 119910(119897) where parameter 119897 isthe position of a given point along the length of the curveThe determination of 119909(119897) and 119910(119897) needs the expansion ofintegrand functions into Maclaurin series and with regard toinverse curvatures into Taylor series

The determination of coordinates of an adequate tran-sition curve in rectangular coordinate system 119909 119910 makesit possible for an easy transfer of it to Gauss-Kruger (119883 119884)

conformal coordinates and its subsequent layout in fieldmaking use of the satellite measuring technique GNSS

The method can be applied to identify various typesof transition curves The paper gives examples of solvingthe problem involving linear change of curvature and thecurvature in polynomial and trigonometric form

The performed comparative analysis reveals a wide rangeof possibilities which are offered by the presented designmethod Its application enables us to generate diversifiedlayout of the route It simultaneously ensures a completecontrol of correctness of the obtained solutions in view ofsatisfying the geometric and kinematic conditions

The presented method of modelling curvature has auniversal character It can be used for vehicular roads andrailway lines as well In the case of railway lines therearises an additional possibility of modelling in a similar waythe superelevation ramp understood here as a determineddifference in the height of the rail courses

Conflict of Interests

The author declares that he has no conflict of interestsregarding the publication of this paper

References

[1] W Koc and C Specht ldquoApplication of the Polish active GNSSgeodetic network for surveying and design of the railroadrdquo inProceedings of the 1st International Conference on Road and RailInfrastructure (CETRA rsquo10) pp 757ndash762 May 2010

[2] P Lipar M Lakner T Maher andM Zura ldquoEstimation of roadcenterline curvature from raw GPS datardquo Baltic Journal of Roadand Bridge Engineering vol 6 no 3 pp 163ndash168 2011

[3] L Tasci and N Kuloglu ldquoInvestigation of a new transitioncurverdquoTheBaltic Journal of Road and Bridge Engineering vol 6no 1 pp 23ndash29 2011

[4] H-H Cai and G-J Wang ldquoA new method in highway routedesign joining circular arcs by a single C-Bezier curve withshape parameterrdquo Journal of Zhejiang University Science A vol10 no 4 pp 562ndash569 2009

[5] S Dimulyo Z Habib and M Sakai ldquoFair cubic transition bet-ween two circles with one circle inside or tangent to the otherrdquoNumerical Algorithms vol 51 no 4 pp 461ndash476 2009

[6] G Farin ldquoClass A Bezier curvesrdquo Computer Aided GeometricDesign vol 23 no 7 pp 573ndash581 2006

[7] Z Habib and M Sakai ldquoG2 pythagorean hodograph quintictransition between two circles with shape controlrdquo ComputerAided Geometric Design vol 24 no 5 pp 252ndash266 2007

[8] G Harary and A Tal ldquoThe natural 3D spiralrdquo Computer Graph-ics Forum vol 30 no 2 pp 237ndash246 2011

[9] K T Miura ldquoA general equation of aesthetic curves and its self-affinityrdquo Computer-Aided Design and Applications vol 3 no 1ndash4 pp 457ndash464 2006

[10] S H Yahaya M S Salleh and J M Ali ldquoSpur gear design withan S-shaped transition curve application using mathematicaand CAD toolsrdquo in Proceedings of the International Conferenceon Computer Technology and Development (ICCTD rsquo09) pp273ndash276 Dubai United Arab Emirates November 2009

[11] N Yoshida and T Saito ldquoQuasi-aesthetic curves in rationalcubic Bezier formsrdquo Computer-Aided Design and Applicationsvol 4 no 1ndash6 pp 477ndash486 2007

[12] R Ziatdinov N Yoshida and T-W Kim ldquoAnalytic parametricequations of log-aesthetic curves in terms of incomplete gammafunctionsrdquo Computer Aided Geometric Design vol 29 no 2 pp129ndash140 2012

[13] R J Grabowski ldquoSmooth curvilinear transitions along vehicu-lar roads and railway linesrdquo Science Papers of AGH Universityof Science and Technology no 82 Cracow Poland 1984(Polish)

[14] L T Klauder ldquoTrack transition curve geometry based on gegen-bauer polynomialsrdquo in Proceedings of the 6th International Con-ference on Railway Engineering May 2003

[15] W Koc Elements of Design Theory on Track Systems GdanskUniversity of Technology Gdansk Poland 2004 (Polish)

[16] E Mieloszyk and W Koc ldquoGeneral dynamic method for det-ermining transition curve equationsrdquo Rail InternationalmdashSchienen der Welt vol 10 pp 32ndash40 1991

[17] B Kufver ldquoRealigning railway in track renewalsmdashlinear versusS-shaped superelevation rampsrdquo in Proceedings of the 2ndInternational Conference on Railway Engineering May 1999

[18] W Koc and E Mieloszyk ldquoComparative analysis of selectedtransition curves by the use of dynamicmodelrdquoArchives of CivilEngineering vol 33 no 2 pp 239ndash261 1987 (Polish)

[19] S Gleason and D Gebre-Egziabher GNSS Applications andMethods Artech House Publishers Norwood NJ USA 2009

Mathematical Problems in Engineering 13

[20] B Hoffmann-Wellenhof H Lichtenegger and E Wasle GNSSGlobal Navigation Satellite SystemsmdashGPS GLONASS Galileo ampMore Springer Wien Austria 2008

[21] P Misra and P Enge Global Positioning System Signals Mea-surements and Performance Ganga-Jamuna Press LincolnMass USA 2012

[22] C Specht ldquoAccuracy and coverage of the modernized PolishMaritime differential GPS systemrdquo Advances in Space Researchvol 47 no 2 pp 221ndash228 2011

[23] W Koc and K Palikowska ldquoDynamic properties evaluation ofthe selected methods of joining route segments with differentcurvaturerdquo Technika Transportu Szynowego vol 9 pp 1785ndash1807 2012 (Polish)

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Complex AnalysisJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

a dynamicmodel In themethod accelerationwas a factor thatexcited the transverse vibrations of the carriage [18]Themainconclusion resulting from the considerations was to provethe relation existing between the response of the system andthe class of the excitation functionThe dynamic interactionswere smaller (ie more advantageous) if the class of thefunction was higher It turned out that evidently the largestacceleration values were noted on the cubical parabola (classof function C0) With respect to Bloss curve and a cosinecurve (class C1) they are significantly smaller whereas onsine curve (class C2) they are the smallest

After years the mentioned identification method ofunbalanced accelerations [16] provided a source of inspira-tion for the elaboration of a new technique for modelling thegeometric system of the communication route based on join-ing two points of the route with diversified curvature

2 Analytical Method ofModelling the Curvature

Ameasure of the route bending is the ratio of the angle whichdetermines the direction of the vehiclersquos longitudinal axis aftercovering a certain arc (Figure 1) The curvature of curve 119870 atpoint119872 is called the boundary which is aimed at by the rela-tion of acute angle ΔΘ between tangents to curve119870 at points119872 and119872

1 to the length of arc Δ119897when point119872

1tends along

curve 119870 to point 119872

119896 = limΔ119897rarr0

1003816100381610038161003816100381610038161003816

ΔΘ

Δ119897

1003816100381610038161003816100381610038161003816=

119889Θ

119889119897 (1)

If the operation procedure by the use of rectangular coor-dinates 119909 119910 is to be continued it is necessary to take into con-sideration the sign of the curvature As illustrated in Figure 1angle ΔΘ = Θ

1minus Θ gt 0 the curvature is assumed here that

it has a positive value which as can be seen accompanies thecurves with downward convexities A negative value of cur-vature corresponds to angle ΔΘ lt 0 and appears on curvesof an upward convexity

Making a generalization of the identification methodapplied to unbalanced accelerations occurring on varioustypes of transition curves [16] it is possible to search for cur-vature function 119896(119897) among the solutions of the differentialequation

119896(119898)

(119897) = 119891 [119897 119896 1198961015840 119896

(119898minus1)] (2)

with conditions for the transition curve at the outset (for 119897 =

0) and at the end (for 119897 = 119897119896)

119896(119894)

(0+) =

1198961

for 119894 = 0

0 for 119894 = 1 2 1198991

119896(119895)

(119897minus

119896) =

1198962

for 119895 = 0

0 for 119895 = 1 2 1198992

(3)

where 1198961and 119896

2indicate the curvature magnitudes on both

ends of the curve (taking note of an adequate sign)The orderof the differential equation (2) is 119898 = 119899

1+ 1198992

+ 2 and

Δx

Δy

x

y

M

K

Δl

Θ1Θ

ΔΘ

ΔΘM1

Figure 1 Schematic diagram for the explanation of the notion ofcurvature

the obtained function 119896(119897) is of class C119899 in the range ⟨0 119897119896⟩

where 119899 = min(1198991 1198992)

The application of themethodunder considerationmakesit possible to combine various geometric elements for exam-ple a straightwith a circular arc and also after introducing anadequate sign of curvature two circular arcs of a consistentrun or converse arcs

On determining the curvature function 119896(119897) a fundamen-tal task is to find the coordinates that correspond to a curve inthe rectangular coordinate system 119909 119910 A solution for such asystem is required by the satellite measuring technique GNSS[19ndash21] which becomes an aid in the determination of thecoordinate points in a uniform 3D-system of coordinatesWGS 84 (the world geodetic system 1984) Next themeasuredellipsoidal coordinates (GPS) are transformed to Gauss-Kruger (119883 119884) conformal coordinates [22] An assumption ismade that the beginning of system119909119910 is at the terminal pointof the input curve (with curvature 119896

1) while the axis of

abscissae is tangent to this curve at this pointThe equation of the sought after connection can be

written in parametric form

119909 (119897) = int cosΘ (119897) 119889119897

119910 (119897) = int sinΘ (119897) 119889119897

(4)

Parameter 119897 is the position of a given point along thecurve length FunctionΘ(119897) is determined on the basis of theformula

Θ (119897) = int 119896 (119897) 119889119897 (5)

The presented method of modelling the curvature has auniversal character It can be applied to both vehicular roadsand railway routes In the case of railway routes there arises anadditional possibility for modelling in a similar way thesuperelevation ramp understood here as a determined differ-ence of heights of the railThe principle of operation is similarto curvaturemodellingThe only differences are that 119896

1has to

be replaced by the value of superelevation ℎ1at the beginning

of the superelevation ramp and 1198962has to be replaced by the

value of superelevation ℎ2at the end of the superelevation

Mathematical Problems in Engineering 3

ramp The ramp length of course corresponds to the lengthof the transition curveUsing the sameprocedure it is possibleto find the unbalanced acceleration 119886(119897)

3 Transition Curve of Linear Curvature

31 Finding Out the Curvature Equation Linear change ofcurvature along a defined length 119897

119896is obtained by assuming

two principal conditions

119896 (0+) = 1198961

119896 (119897minus

119896) = 1198962

(6)

and a differential equation

11989610158401015840(119897) = 0 (7)

After the determination of constants the solution of thedifferential problem (6) (7) is as follows

119896 (119897) = 1198961+

1

119897119896

(1198962minus 1198961) 119897 (8)

Function Θ(119897) is obtained from formula (5) In the caseunder consideration

Θ (119897) = 1198961119897 +

1

2119897119896

(1198962minus 1198961) 1198972 (9)

32 The Determination of the Transition Curve Length (forRailway Lines) With regard to rail tracks the transition curvemust satisfy two kinematic conditions

120595max le 120595dop (10)

119891max le 119891dop (11)

where 120595max is the maximum value of acceleration incrementon transition curve inms3120595dop is permissible value of accel-eration increment in ms3 119891max is the maximum speed oflifting the wheel on superelevation ramp in mms and 119891dop ispermissible value of speed of lifting the wheel in mms

The unbalanced acceleration 119886(119897) occurring on the tran-sition curve results from speed V of the train magnitude ofcurvature and ordinates of the superelevation ramp itscourse is similar to 119896(119897) and ℎ(119897) Analogously with (8)

119886 (119897) = 1198861+

1

119897119896

(1198862minus 1198861) 119897 (12)

where 1198861and 119886

2are accelerations at ends of the transition

curve

1198861=

V2

rad1198961minus 119892

ℎ1

119904 119886

2=

V2

rad1198962minus 119892

ℎ2

119904

(13)

(ℎ1and ℎ

2are ordinates of the superelevation ramp at its

ends)On the assumption that the speed of the train V = const

the formula for acceleration increment 120595(119897) is as follows

120595 (119897) = V119889

119889119897119886 (119897) =

V119897119896

(1198862minus 1198861) (14)

As can be seen the acceleration increment is here a con-stant value However from condition (10) it follows that

119897120595

119896ge

V120595dop

10038161003816100381610038161198862 minus 1198861

1003816100381610038161003816 (15)

The superelevation ramp equation in the case under con-sideration is similar to (8) and (12)

ℎ (119897) = ℎ1+

1

119897119896

(ℎ2minus ℎ1) 119897 (16)

The speed of lifting the wheel 119891(119897) on the superelevationramp (assuming that the travelling speed of the train V =const) is determined by the equation

119891 (119897) = V119889

119889119897ℎ (119897) =

V119897119896

(ℎ2minus ℎ1) (17)

Thus also119891(119897) is here a constant value but fromcondition(11) it follows that

119897119891

119896ge

V119891dop

1003816100381610038161003816ℎ2 minus ℎ1

1003816100381610038161003816 (18)

The assumed length of the transition curvemust fulfill thecondition

119897119896ge max (119897

120595

119896 119897119891

119896) (19)

33 Coordinates of Transition Curve Connecting Uniform Cur-vatures The determination of 119909(119897) and 119910(119897) by using (4) willneed the expansion of integrands intoMaclaurin series Aftercompleting the whole procedure the following parametricequations are obtained

119909 (119897) = int cosΘ (119897) 119889119897

= 119897 minus1198962

1

61198973minus

1198961

8119897119896

(1198962minus 1198961) 1198974

+ [1198964

1

120minus

1

401198972

119896

(1198962minus 1198961)2] 1198975+

1198963

1

72119897119896

(1198962minus 1198961) 1198976

minus [1198966

1

5040minus

1198962

1

1121198972

119896

(1198962minus 1198961)2] 1198977

minus [1198965

1

1920119897119896

(1198962minus 1198961) minus

1198961

3841198973

119896

(1198962minus 1198961)3] 1198978

+ [1198968

1

362880minus

1198964

1

17281198972

119896

(1198962minus 1198961)2

+1

34561198974

119896

(1198962minus 1198961)4] 1198979sdot sdot sdot

4 Mathematical Problems in Engineering

119910 (119897) = int sinΘ (119897) 119889119897

=1198961

21198972+

1

6119897119896

(1198962minus 1198961) 1198973minus

1198963

1

241198974minus

1198962

1

20119897119896

(1198962minus 1198961) 1198975

+ [1198965

1

720minus

1198961

481198972

119896

(1198962minus 1198961)2] 1198976

+ [1198964

1

336119897119896

(1198962minus 1198961) minus

1

3361198973

119896

(1198962minus 1198961)3] 1198977

minus [1198967

1

40320minus

1198963

1

3841198972

119896

(1198962minus 1198961)2] 1198978

minus [1198966

1

12960119897119896

(1198962minus 1198961) minus

1198962

1

8641198973

119896

(1198962minus 1198961)3] 1198979+ sdot sdot sdot

(20)

Equations (20) for 1198961= 0 describe the curve in the form

of clothoid used to connect a straight with a circular arc

34 Coordinates of Transition Curve Connecting Inverse Cur-vatures At inverse arcs (ie with diversified curvature signs)there arises the problem related to the determined coordi-nates 119909(119897) and 119910(119897) due to the course of function Θ(119897) Forsign 119896

1= sign 119896

2function Θ(119897) is a monotonic function

whereas for sign 1198961

= sign 1198962in the diagram of function Θ(119897)

there appears an extremum at point 1198970 where

Θ1015840(1198970) = 119896 (119897

0) = 1198961+ (1198962minus 1198961)1198970

119897119896

= 0 (21)

Values 1198970and Θ(119897

0) follow from the relations

1198970= minus

1198961

1198962minus 1198961

119897119896

Θ(1198970) = minus

1198962

1

2 (1198962minus 1198961)119897119896

(22)

Figure 2 illustrates a scheme of angle Θ(119897) for the con-nection of inverse circular arcs of radii 119877

1= 1200m and

1198772= 700mThe length of the transition curve 119897

119896= 60m was

determined for speed value V = 90 kmh using the procedureas described at Section 32 (the assumed superelevationvalues were ℎ

1= 20mm and ℎ

2= 45mm)

In this situation the parametric equations of transitioncurve (20) are effective for 119897 isin ⟨0 119897

0⟩ for 119897 isin (119897

0 119897119896⟩ functions

0

0005

001

0015

0 20 40 60 80minus0005

minus001

minus0015

minus002

Q(rad)

l (m)

Figure 2 Diagram of angle Θ(119897) defined by (9) for circulararcs connection of curvatures 119896

1= 11200 radm and 119896

2=

minus1700 radm by means of transition curve of length 119897119896= 60m

cosΘ(119897) and sinΘ(119897) should be expanded to Taylor seriesAfter integration of the equations take the following form

119909 (119897) = int cosΘ (119897) 119889119897

= 119909 (1198970) + cos(minus

1

2

1198962

1

1198962minus 1198961

119897119896) (119897 minus 119897

0)

minus [1

6119897119896

(1198962minus 1198961) sin(minus

1

2

1198962

1

1198962minus 1198961

119897119896)] (119897 minus 119897

0)3

minus [1

401198972

119896

(1198962minus 1198961)2 cos(minus

1

2

1198962

1

1198962minus 1198961

119897119896)] (119897 minus 119897

0)5

+ [1

3361198973

119896

(1198962minus 1198961)3 sin(minus

1

2

1198962

1

1198962minus 1198961

119897119896)] (119897 minus 119897

0)7

+ sdot sdot sdot

(23)

119910 (119897) = int sinΘ (119897) 119889119897

= 119910 (1198970) + sin(minus

1

2

1198962

1

1198962minus 1198961

119897119896) (119897 minus 119897

0)

+ [1

6119897119896

(1198962minus 1198961) cos(minus

1

2

1198962

1

1198962minus 1198961

119897119896)] (119897 minus 119897

0)3

minus [1

401198972

119896

(1198962minus 1198961)2 sin(minus

1

2

1198962

1

1198962minus 1198961

119897119896)] (119897 minus 119897

0)5

minus [1

3361198973

119896

(1198962minus 1198961)3 cos(minus

1

2

1198962

1

1198962minus 1198961

119897119896)] (119897 minus 119897

0)7

+ [1

34561198974

119896

(1198962minus 1198961)4 sin(minus

1

2

1198962

1

1198962minus 1198961

119897119896)] (119897 minus 119897

0)9

+ sdot sdot sdot

(24)

Mathematical Problems in Engineering 5

On completing the parametric equations (20) and (23)along with (24) a correct solution of the task for the caseof diversified curvature signs is obtained (Figure 3) From apractical point of view the determination of the magnitude ofthe tangent slope angle at the end of the transition curve isvery important It amounts to

Θ(119897119896) =

1

2(1198961+ 1198962) 119897119896 (25)

A knowledge of Θ(119897119896) makes it possible to add the tran-

sition curve to the other one having curvature 1198962at its initial

point (satisfying the tangency condition of both the curves)

4 Transition Curve of Polynomial Curvature

From the point of view of vehiclesrsquo dynamics the transitioncurve of linear change of curvature is not the most advan-tageous solution [23] A definitely better one is nonlinearsolution whose characteristics are interesting enough to becomparedwith the Bezier curves preferable lately in literature[4 7]

The differential equation (2) allows to appoint an unlim-ited number of curves with nonlinear curvature change Inthe case of geometric layout of communication routes it willbe appropriate to consider the transition curves of poly-nomial and trigonometric curvature In fact such forms ofcurves are used in the standard ways to connect a straight linewith a circular arc

41 Formulation of the Curvature Equation To formulate thecurvature equation in the polynomial form let us assume thefollowing boundary conditions

119896 (0+) = 1198961 119896 (119897

minus

119896) = 1198962

1198961015840(0+) = 0 119896

1015840(119897minus

119896) = 0

(26)

and the differential equation

119896(4)

(119897) = 0 (27)

After solving the differential problem (26) (27) thefollowing formula for curvature 119896(119897) is obtained

119896 (119897) = 1198961+

3

1198972

119896

(1198962minus 1198961) 1198972minus

2

1198973

119896

(1198962minus 1198961) 1198973 (28)

The form of function Θ(119897) determined on the basis of (5)and (28) is as follows

Θ (119897) = 1198961119897 +

1

1198972

119896

(1198962minus 1198961) 1198973minus

1

21198973

119896

(1198962minus 1198961) 1198974 (29)

On rail routes the length of the transition curve isdetermined like at Section 32 for solution of linear curvatureThe following conditions should be fulfilled here

119897120595

119896ge

3

2

V120595dop

10038161003816100381610038161198862 minus 1198861

1003816100381610038161003816

119897119891

119896ge

3

2

V119891dop

1003816100381610038161003816ℎ2 minus ℎ1

1003816100381610038161003816

(30)

0

01

02

03

04

0 20 40 60 80x (m)

y(m

)

Figure 3 Diagram of horizontal ordinates 119910(119909) defined by (20)(23) and (24) for the joint of circular arcs of curvatures 119896

1=

11200 radm and 1198962

= minus1700 radm with transition curve oflength 119897

119896= 60m (using different horizontal and vertical scales)

The adopted length of the transition curve must satisfycondition (19)

42 Coordinates of Transition Curve Connecting Uniform Cur-vatures On expanding functions cosΘ(119897) and sinΘ(119897) intoMaclaurin series and after integration the following paramet-ric equations are acquired

119909 (119897) = int cosΘ (119897) 119889119897

= 119897 minus1198962

1

61198973+ [

1198964

1

120minus

1198961

51198972

119896

(1198962minus 1198961)] 1198975

+ [1198961

121198973

119896

(1198962minus 1198961)] 1198976

minus [1198966

1

5040minus

1198963

1

421198972

119896

(1198962minus 1198961) +

1

141198974

119896

(1198962minus 1198961)2] 1198977

minus [1198963

1

961198973

119896

(1198962minus 1198961) minus

1

161198975

119896

(1198962minus 1198961)2] 1198978

+ [1198968

1

362880minus

1198965

1

10801198972

119896

(1198962minus 1198961) +

1198962

1

361198974

119896

(1198962minus 1198961)2

+1

721198976

119896

(1198962minus 1198961)2] 1198979+ sdot sdot sdot

119910 (119897) = int sinΘ (119897) 119889119897

=1198961

21198972minus [

1198963

1

24minus

1

41198972

119896

(1198962minus 1198961)] 1198974

minus1

101198973

119896

(1198962minus 1198961) 1198975+ [

1198965

1

720minus

1198962

1

121198972

119896

(1198962minus 1198961)] 1198976

+1198962

1

281198973

119896

(1198962minus 1198961) 1198977

6 Mathematical Problems in Engineering

0

00005

0001

0 20 40 60 80 100l (m)minus00005

minus0001

minus00015

minus0002

k(radm

)

Figure 4 Diagram of curvature 119896(119897) described by (28) for theconnection of circular arcs of curvatures 119896

1= 11200 radm and

1198962

= minus1700 radm with the use of transition curve of length 119897119896

=

90m

minus [1198967

1

40320minus

1198964

1

1921198972

119896

(1198962minus 1198961) +

1198961

161198974

119896

(1198962minus 1198961)2] 1198978

+ [1198961

181198975

119896

(1198962minus 1198961)2minus

1198964

1

4321198973

119896

(1198962minus 1198961)] 1198979+ sdot sdot sdot

(31)

Equations (31) for 1198961

= 0 describe the so-called Blosscurve which can be used to join a straight to a circular arc

43 Coordinates of Transition Curve Connecting Inverse Cur-vatures When the arcs are inverse (ie the curvature signsare different Figure 4) there occurs a problem relating to thedetermined coordinates 119909(119897) and 119910(119897) which results fromfunction Θ(119897) For sign 119896

1= sign 119896

2function Θ(119897) is a mono-

tonic function while for sign 1198961

= sign 1198962in the diagram of

function Θ(119897) there appears extremum at point 1198970 where

Θ1015840(1198970) = 119896 (119897

0) = 1198961+

3

1198972

119896

(1198962minus 1198961) 1198972

0minus

2

1198973

119896

(1198962minus 1198961) 1198973

0= 0

(32)

Parametric equations of transition curve (31) are valid for119897 isin ⟨0 119897

0⟩The determination of value 119897

0makes it necessary to

solve (32) which is cubic The sought-after value 1198970

isin ⟨0 119897119896⟩

satisfying conditions of the problem is defined by the formula

1198970= [

1

2minus cos(

120593

3+

120587

3)] 119897119896 (33)

where angle 120593 follows from the relation

cos120593 =1198961+ 1198962

1198962minus 1198961

(34)

Under such circumstances in order to determine para-metric equations of the transition curve for 119897 isin (119897

0 119897119896⟩ func-

tions cosΘ(119897) and sinΘ(119897) should be expanded into Taylor

series after integration the following parametric equations ofthe transition curve are obtained

119909 (119897) = int cosΘ (119897) 119889119897

= 119909 (1198970) + cosΘ(119897

0) (119897 minus 119897

0)

minus1

6sinΘ(119897

0)Θ10158401015840(1198970) (119897 minus 119897

0)3

minus1

24sinΘ(119897

0)Θ101584010158401015840

(1198970) (119897 minus 119897

0)4

minus 1

40cosΘ(119897

0) [Θ10158401015840(1198970)]2

+1

120sinΘ(119897

0)Θ(4)

(1198970) (119897 minus 119897

0)5

minus1

72cosΘ(119897

0)Θ10158401015840(1198970)Θ101584010158401015840

(1198970) (119897 minus 119897

0)6

+ 1

336sinΘ(119897

0) [Θ10158401015840(1198970)]3

minus1

336cosΘ(119897

0)Θ10158401015840(1198970)Θ(4)

(1198970)

minus1

504cosΘ(119897

0) [Θ101584010158401015840

(1198970)]2

(119897 minus 1198970)7

+ 1

384sinΘ(119897

0) [Θ10158401015840(1198970)]2

Θ101584010158401015840

(1198970)

minus1

1152cosΘ(119897

0)Θ101584010158401015840

(1198970)Θ(4)

(1198970) (119897 minus 119897

0)8

+ 1

3456cosΘ(119897

0) [Θ10158401015840(1198970)]4

+1

1296sinΘ(119897

0)Θ10158401015840(1198970) [Θ101584010158401015840

(1198970)]2

+1

1728sinΘ(119897

0) [Θ10158401015840(1198970)]2

Θ(4)

(1198970)

minus1

10368cosΘ(119897

0) [Θ(4)

(1198970)]2

(119897 minus 1198970)9+ sdot sdot sdot

119910 (119897) = int sinΘ (119897) 119889119897

= 119910 (1198970) + sinΘ(119897

0) (119897 minus 119897

0)

+1

6[cosΘ(119897

0)Θ10158401015840(1198970)] (119897 minus 119897

0)3

+1

24[cosΘ(119897

0)Θ101584010158401015840

(1198970)] (119897 minus 119897

0)4

minus 1

40sinΘ(119897

0) [Θ10158401015840(1198970)]2

minus1

120cosΘ(119897

0)Θ(4)

(1198970) (119897 minus 119897

0)5

minus1

72[sinΘ(119897

0)Θ10158401015840(1198970)Θ101584010158401015840

(1198970)] (119897 minus 119897

0)6

Mathematical Problems in Engineering 7

minus 1

336cosΘ(119897

0) [Θ10158401015840(1198970)]3

+1

336sinΘ(119897

0)Θ10158401015840(1198970)Θ(4)

(1198970)

+1

504sinΘ(119897

0) [Θ101584010158401015840

(1198970)]2

(119897 minus 1198970)7

minus 1

384cosΘ(119897

0) [Θ10158401015840(1198970)]2

Θ101584010158401015840

(1198970)

+1

1152sinΘ(119897

0)Θ101584010158401015840

(1198970)Θ(4)

(1198970) (119897 minus 119897

0)8

+ 1

3456sinΘ(119897

0) [Θ10158401015840(1198970)]4

minus1

1296cosΘ(119897

0)Θ10158401015840(1198970) [Θ101584010158401015840

(1198970)]2

minus1

1728cosΘ(119897

0) [Θ10158401015840(1198970)]2

Θ(4)

(1198970)

minus1

10368sinΘ(119897

0) [Θ(4)

(1198970)]2

(119897 minus 1198970)9+ sdot sdot sdot

(35)

where

Θ(1198970) = 11989611198970+

1

1198972

119896

(1198962minus 1198961) 1198973

0minus

1

21198973

119896

(1198962minus 1198961) 1198974

0

Θ10158401015840(1198970) =

6

1198972

119896

(1198962minus 1198961) 1198970minus

6

1198973

119896

(1198962minus 1198961) 1198972

0

Θ101584010158401015840

(1198970) =

6

1198972

119896

(1198962minus 1198961) minus

12

1198973

119896

(1198962minus 1198961) 1198970

Θ(4)

(1198970) = minus

12

1198973

119896

(1198962minus 1198961)

(36)

Making use of parametric equations (31) and also (35) onecan obtain a correct solution of the problem with diversifiedsigns of curvature

Value of the tangent slope at the end of the transitioncurvemakes it possible to connect to it a curve of curvature 119896

2

satisfying simultaneously the tangency condition of both thecurves The above value is the same as in the case of linearcurvature It is defined by (25)

5 Transition Curve of Curvature inTrigonometric Form

51 Formulation of Curvature Equation As before boundaryconditions (26) and another differential equation areassumed

119896(4)

(119897) +1205872

1198972

119896

11989610158401015840(119897) = 0 (37)

As soon as the differential problem (26) (37) has beensolved a formula for curvature is obtained

119896 (119897) =1

2(1198961+ 1198962) minus

1

2(1198962minus 1198961) cos 120587

119897119896

119897 (38)

The form of function Θ(119897) determined by the use of (5)and (38) is as follows

Θ (119897) =1

2(1198961+ 1198962) 119897 minus

1

2(1198962minus 1198961)119897119896

120587sin 120587

119897119896

119897 (39)

The length of the transition curve on railway routesfollows from conditions

119897120595

119896ge

120587

2

V120595dop

10038161003816100381610038161198862 minus 1198861

1003816100381610038161003816

119897119891

119896ge

120587

2

V119891dop

1003816100381610038161003816ℎ2 minus ℎ1

1003816100381610038161003816

(40)

The assumed length of the curve must satisfy the condi-tion (19)

52 Ordinates of Transition Curve Connecting UniformCurva-tures After the expansion of functions cosΘ(119897) and sinΘ(119897)

intoMaclaurin series to be next integrated the following para-metric equations are obtained

119909 (119897) = int cosΘ (119897) 119889119897

= 119897 minus1198962

1

61198973+ [

1198964

1

120minus

1

60

1205872

1198972

119896

1198961(1198962minus 1198961)] 1198975

minus [1198966

1

5040minus

1

504

1205872

1198972

119896

1198963

1(1198962minus 1198961)

minus1

1680

1205874

1198974

119896

1198961(1198962minus 1198961) minus

1

2016

1205874

1198974

119896

(1198962minus 1198961)2] 1198977

+ [1198968

1

362880minus

1

12960

1205872

1198972

119896

1198965

1(1198962minus 1198961)

minus1

5184

1205874

1198974

119896

1198962

1(1198962minus 1198961)2

minus1

12960

1205874

1198974

119896

1198963

1(1198962minus 1198961) minus

1

90720

1205876

1198976

119896

1198961(1198962minus 1198961)

+1

25920

1205876

1198976

119896

(1198962minus 1198961)2] 1198979+ sdot sdot sdot

8 Mathematical Problems in Engineering

119910 (119897) = int sinΘ (119897) 119889119897

=1

211989611198972minus [

1

241198963

1minus

1

48

1205872

1198972

119896

(1198962minus 1198961)] 1198974

+ [1

7201198965

1minus

1

1441198962

1

1205872

1198972

119896

(1198962minus 1198961)

minus1

1440

1205874

1198974

119896

(1198962minus 1198961)] 1198976

minus [1

403201198967

1minus

1

23041198964

1

1205872

1198972

119896

(1198962minus 1198961)

+1

2304

1205874

1198974

119896

1198961(1198962minus 1198961)2minus

1

3840

1205874

1198974

119896

1198962

1(1198962minus 1198961)

+1

80640

1205876

1198976

119896

(1198962minus 1198961)] 1198978+ sdot sdot sdot

(41)

Equations (41) for 1198961

= 0 describe the curve known ascosinusoid which can be used to connect a straight with acircular arc

53 Transition Curve Coordinates Connecting Inverse Curva-tures Near inverse arcs in the diagram of functionΘ(119897) thereappears the extremum at point 119897

0 where

Θ1015840(1198970) = 119896 (119897

0) =

1

2(1198961+ 1198962) minus

1

2(1198962minus 1198961) cos 120587

119897119896

1198970= 0

(42)

The transition curve parametric equations (41) are effec-tive for 119897 isin ⟨0 119897

0⟩ To determine value 119897

0one must solve (42)

The equation can be written in the form

cos 120587

119897119896

1198970=

1198961+ 1198962

1198962minus 1198961

(43)

It is easy to prove that for sign 1198961

= sign 1198962it is necessary

to satisfy the condition |1198961+1198962| lt |1198962minus1198961| thus |(119896

1+1198962)(1198962minus

1198961)| lt 1 The formula for length 119897

0is as follows

1198970=

1

120587(arccos1198961 + 119896

2

1198962minus 1198961

) 119897119896 (44)

For 119897 isin (1198970 119897119896⟩ functions cosΘ(119897) and sinΘ(119897) should

be expanded into Taylor series The integration provides thefollowing equations

119909 (119897) = int cosΘ (119897) 119889119897

= 119909 (1198970) + cosΘ(119897

0) (119897 minus 119897

0)

minus1

6sinΘ(119897

0)Θ10158401015840(1198970) (119897 minus 119897

0)3

minus1

24sinΘ(119897

0)Θ101584010158401015840

(1198970) sdot (119897 minus 119897

0)4

minus 1

40cosΘ(119897

0) [Θ10158401015840(1198970)]2

+1

120sinΘ(119897

0)Θ(4)

(1198970) (119897 minus 119897

0)5

minus 1

72cosΘ(119897

0)Θ10158401015840(1198970)Θ101584010158401015840

(1198970)

+1

720sinΘ(119897

0)Θ(5)

(1198970) (119897 minus 119897

0)6

+ 1

336sinΘ(119897

0) [Θ10158401015840(1198970)]3

minus1

336cosΘ(119897

0)Θ10158401015840(1198970)Θ(4)

(1198970)

minus1

504cosΘ(119897

0) [Θ101584010158401015840

(1198970)]2

minus1

5040sinΘ(119897

0)Θ(6)

(1198970) (119897 minus 119897

0)7

+ 1

384sinΘ(119897

0) [Θ10158401015840(1198970)]2

Θ101584010158401015840

(1198970)

minus1

1920cosΘ(119897

0)Θ10158401015840(119897) Θ(5)

(119897)

minus1

1152cosΘ(119897

0)Θ101584010158401015840

(1198970)Θ(4)

(1198970)

minus1

40320sinΘ(119897

0)Θ(7)

(1198970) (119897 minus 119897

0)8

+ 1

3456cosΘ(119897

0) [Θ10158401015840(1198970)]4

+1

1296sinΘ(119897

0)Θ10158401015840(1198970) [Θ101584010158401015840

(1198970)]2

+1

1728sinΘ(119897

0) [Θ10158401015840(1198970)]2

Θ(4)

(1198970)

minus1

12960cosΘ(119897

0)Θ10158401015840(1198970)Θ(6)

(1198970)

minus1

6480cosΘ(119897

0)Θ101584010158401015840

(1198970)Θ(5)

(1198970)

minus1

10368cosΘ(119897

0) [Θ(4)

(1198970)]2

minus1

362880sinΘ(119897

0)Θ(8)

(1198970) (119897 minus 119897

0)9+ sdot sdot sdot

Mathematical Problems in Engineering 9

119910 (119897) = int sinΘ (119897) 119889119897

= 119910 (1198970) + sinΘ(119897

0) (119897 minus 119897

0)

+1

6cosΘ(119897

0)Θ10158401015840(1198970) (119897 minus 119897

0)3

+1

24cosΘ(119897

0)Θ101584010158401015840

(1198970) sdot (119897 minus 119897

0)4

minus 1

40sinΘ(119897

0) [Θ10158401015840(1198970)]2

minus1

120cosΘ(119897

0)Θ(4)

(1198970) (119897 minus 119897

0)5

minus 1

72sinΘ(119897

0) 11987610158401015840(1198970)Θ101584010158401015840

(1198970)

minus1

720cosΘ(119897

0)Θ(5)

(1198970) (119897 minus 119897

0)6

minus 1

336cosΘ(119897

0) [Θ10158401015840(1198970)]3

+1

336sinΘ(119897

0)Θ10158401015840(1198970)Θ(4)

(1198970)

+1

504sinΘ(119897

0) [Θ101584010158401015840

(1198970)]2

minus1

5040cosΘ(119897

0)Θ(6)

(1198970) (119897 minus 119897

0)7

minus 1

384cosΘ(119897

0) [Θ10158401015840(1198970)]2

Θ101584010158401015840

(1198970)

+1

1920sinΘ(119897

0)Θ10158401015840(1198970)Θ(5)

(1198970)

+1

1152sinΘ(119897

0)Θ101584010158401015840

(1198970)Θ(4)

(1198970)

minus1

40320cosΘ(119897

0)Θ(7)

(1198970) (119897 minus 119897

0)8

+ 1

3456sinΘ(119897

0) [Θ10158401015840(1198970)]4

minus1

1296cosΘ(119897

0)Θ10158401015840(1198970) [Θ101584010158401015840

(1198970)]2

minus1

1728cosΘ(119897

0) [Θ10158401015840(1198970)]2

Θ(4)

(1198970)

minus1

12960sin119876 (119897

0) 11987610158401015840(1198970) 119876(6)

(1198970)

minus1

6480sin119876 (119897

0) 119876101584010158401015840

(1198970) 119876(5)

(1198970)

minus1

10368sinΘ(119897

0) [Θ(4)

(1198970)]2

+1

362880cosΘ(119897

0)Θ(8)

(1198970) (119897 minus 119897

0)9+ sdot sdot sdot

(45)

where

Θ(1198970) =

1

2(1198961+ 1198962) 1198970plusmn

119897119896

120587radicminus11989611198962

Θ10158401015840(1198970) = ∓

120587

119897119896

radicminus11989611198962

Θ10158401015840(1198970) = ∓

120587

119897119896

radicminus11989611198962

Θ101584010158401015840

(1198970) =

1

2

1205872

1198972

119896

(1198961+ 1198962)

Θ(4)

(1198970) = plusmn

1205873

1198973

119896

radicminus11989611198962

Θ(5)

(1198970) = minus

1

2

1205874

1198974

119896

(1198961+ 1198962)

Θ(6)

(1198970) = ∓

1205875

1198975

119896

radicminus11989611198962

Θ(7)

(1198970) =

1

2

1205876

1198976

119896

(1198961+ 1198962)

Θ(8)

(1198970) = plusmn

1205877

1198977

119896

radicminus11989611198962

(46)

In expressions with double signs plusmn and ∓ the upper onecorresponds to the case 119896

1gt 0 119896

2lt 0 while the lower one

refers to 1198961lt 0 119896

2gt 0

The value of tangent slope at the end of the transitioncurvemakes it possible to connect to it a curve of curvature 119896

2

satisfying the tangency condition of both the curves Thevalue is the same as in the previous cases It is determinedby (25)

6 Comparison of Horizontal Ordinates ofDetermined Transition Curves

61 The Shaping of Horizontal Ordinates of ComparableCurves If the analysis is carried out for railway routes thenthe necessity of meeting kinematic conditions (10) and (11)will require a diversification of the lengths of the comparedwith each other curves (using the same speed of trains)Moreover it is necessary to individually consider cases ofconnecting homogeneous curvatures (sign 119896

1= sign 119896

2) and

inverse arcs (sign 1198961

= sign 1198962)

It is assumed that at joints of consistent arcs with radii1198771= 1200m and 119877

2= 700m an attempt is made to attain a

speed V = 110 kmh Hence it is necessary to employ on arcssuperelevation of ℎ

1= 80mmand ℎ

2= 115mm respectively

The retention of the same kinematic parameters can providediversified lengths of the transition curve

(i) for a curve of linear curvature 119897119896= 40m

(ii) for a curve of polynomial curvature 119897119896= 60m

(iii) for a curve of trigonometric curvature 119897119896= 62 832m

10 Mathematical Problems in Engineering

The above length relations result directly from the equa-tions given at Sections 32 41 and 51

Figure 5 gives diagrams of horizontal ordinates for com-pared transition curves of positive and negative curvaturevalues As seen the curvature ordinates of linear curvatureevidently deviate from the ordinates of other curves In turncurves of polynomial and trigonometric curvature are inprinciple consistent A difference can only be noted in thefinal area And so the class of the function describing thecurvature plays here a significant part with respect to a curveof linear curvature it is a function of class C0 while regardingthe other two curves it is a function of class C1

At the joint of inverse arcs of radii 1198771= 1200m and 119877

2=

700m an attempt is made to reach the speed of V = 90 kmhHence it follows that it is necessary to apply superelevationalong arcs attaining ℎ

1= 20mm and ℎ

2= 45mm respec-

tively By the use the same kinematic parameters it is possibleto obtain the following lengths of the transition curve

(i) for a curve of linear curvature 119897119896= 60m

(ii) for a curve of polynomial curvature 119897119896= 90m

(iii) for a curve of trigonometric curvature 119897119896= 94 248m

Figure 6 presents diagrams of horizontal ordinates ofcompared transition curves for positive and negative valuesof curvature

As can be seen like in the case of consistent arcs the curveordinates of linear curvature deviate significantly from theordinates of the remaining curves but the shape of thesecurves also differs one from the other The class of functiondescribing the curvature is still important However in therange of the same class various geometric solutions arepossible

62 Negotiation of Transition Curves in a Geometric SystemThe transition curves do not appear alone in a geometricsystem but must be connected with some arcs in the neigh-borhood If the input arc is circular then its equation is asfollows

(i) for 1198961gt 0

119910 (119909) = 1198771minus radic1198772

1minus 1199092 (47)

(ii) for 1198961lt 0

119910 (119909) = minus (1198771minus radic1198772

1minus 1199092) (48)

where 119909 lt 0

However if the output arc is also a circular arc then it isexpressed in the following equation

(i) for 1198962gt 0

119910 (119909) = 119910119878minus radic1198772

2minus (119909119878minus 119909)2

119909119878= 119909119870

minus119904119870

radic1 + 1199042

119870

1198772 119910

119878= 119910119870

+1

radic1 + 1199042

119870

1198772

(49)

0051

152

25

0 10 20 30 40 50 60 70x (m)

y(m

)

minus05

minus1

minus15

minus2

minus25

Figure 5 Diagrams of horizontal ordinates of curves joiningconsistent arcs for positive and negative values of curvature (usingdifferent horizontal and vertical scales) curve of linear curvatureline in violet colour curve of polynomial curvature in green colourand curve of trigonometric curvature in red colour

0

05

1

15

0 20 40 60 80 100x (m)

y(m

)

minus05

minus1

minus15

Figure 6 Diagrams of horizontal ordinates of curves connectinginverse arcs for positive and negative values of curvature (usingdifferent horizontal and vertical scales) the curve of linear curvatureis in violet color the curve of polynomial curvature is in green colorand the curve of trigonometric curvature is in red color

(ii) for 1198962lt 0

119910 (119909) = 119910119878+ radic1198772

2minus (119909119878minus 119909)2

119909119878= 119909119870

+119904119870

radic1 + 1199042

119870

1198772 119910

119878= 119910119870

minus1

radic1 + 1199042

119870

1198772

(50)

where 119909 gt 119909119870

In (49) and (50) 119909119870denotes the abscissae of the transition

curve end119910119870is the ordinate of the transition curve endwhile

119904119870is the value of tangent at the end of the transition curve

Values for numerical examples under consideration are givenin Table 1

Figures 7 and 8 illustrate the geometric systems made upof two circular arcs of consistent curvature connected withvarious types of transition curves As seen the application of agiven type of transition curve leads to an adequate position ofthe output circular arcThe position of this arc by the use of alinear transition curve clearly deviates from such cases whereadvantage is taken of curves with polynomial and trigono-metrical curvatureWith regard to the latter amutual positionof the circular arc varies insignificantly

Mathematical Problems in Engineering 11

Table 1 Comparison of values 119909119870 119910119870 and 119904

119870for numerical examples analyzed

Curve 119897119896[m] Θ(119897

119896) [rad] 119904

119870119909119870[m] 119910

119870[m]

Case 1 1198961= 11200 radm and 119896

2= 1700 radm

Curve C0 40 0045238 0045269 39988 0825Curve C1 (p) 60 0067857 0067961 59963 1821Curve C1 (t) 62832 0071060 0071180 62796 1940

Case 2 1198961= 11200 radm and 119896

2= minus1700 radm

Curve C0 60 minus0017857 minus0017859 59999 0143Curve C1 (p) 90 minus0026786 minus0026792 89991 0627Curve C1 (t) 94248 minus0028050 minus0028057 94238 0714

Case 3 1198961= minus11200 radm and 119896

2= 1700 radm

Curve C0 60 001786 0017859 59999 minus0143Curve C1 (p) 90 0026786 0026792 89991 minus0627Curve C1 (t) 94248 0028050 0028057 94238 minus0714

Case 4 1198961= minus11200 radm and 119896

2= minus1700 radm

Curve C0 40 minus0045238 minus0045269 39988 minus0825Curve C1 (p) 60 minus0067857 minus0067961 59969 minus1821Curve C1 (t) 62832 minus0071060 minus0071180 62976 minus1940

0

1

2

3

4

5

6

0 20 40 60 80 100 120x (m)

y(m

)

minus40 minus20

Figure 7 Examples of geometric systems consisting of two circulararcs with consistent curvature (brown line) connected with differenttransition curves for positive values of curvature (using differenthorizontal and vertical scales) curve of linear curvature in violetline curve of polynomial curvature in green and curve of trigono-metric curvature in red

00 20 40 60 80 100 120

x (m)

y(m

)

minus40 minus20

minus1

minus2

minus3

minus4

minus5

minus6

minus7

Figure 8 Examples of geometric systems consisting of two circulararcs with consistent curvature (brown line) connected with differenttransition curves for negative values of curvature (using differenthorizontal and vertical scales) curve of linear curvature in violetline curve of polynomial curvature in green and curve of trigono-metric curvature in red

0

05

1

15

0 50 100 150x (m)

y(m

)

minus05

minus1

minus15

minus2

minus50

Figure 9 Examples of geometric systems consisting of two inversecircular arcs (brown line) connected with different types of transi-tion curves for positive values of input curvature (using differenthorizontal and vertical scales) curve of linear curvature in violetline curve of polynomial curvature in green and curve of trigono-metric curvature in red

Figures 9 and 10 illustrate geometric systems made up oftwo inverse circular arcs connected with various types oftransition curves

Here the layout of the output circular arc is much morediversifiedThe position of the arc obtained by the applicationof the transition curve of linear curvature diverts by severaldozenmeters from situationswhere curves of polynomial andtrigonometric curvature have been used The difference forcurves of polynomial and trigonometric curvature attains inthe numerical example analyzed an order of several meters

The presented examples are taken in the very nature ofthings at random They reveal wide possibilities for the pre-sented analytical method in modelling geometric systems Itsapplication makes it possible to generate diversified layout ofthe communication routes adapted to the field conditions andother circumstances Simultaneously the method ensurescorrectness of the obtained solution in view of the kinematicparameters magnitudes in force

12 Mathematical Problems in Engineering

0

05

1

15

2

0 50 100 150x (m)

y(m

)

minus05

minus1

minus15

minus50

Figure 10 Examples of geometric systems consisting of twoinverse circular arcs (brown line) connected with different typesof transition curves for negative values of input curvature (usingdifferent horizontal and vertical scales) curve of linear curvaturein violet line curve of polynomial curvature in green and curve oftrigonometric curvature in red

7 Conclusions

The presented universal method of modelling geometricalsystem of a communication route is based on the determina-tion of an adequate curvature equation by using differentialequationsThe technique enables us to design connections ofvarious geometrical elements of the route indicating diversifi-cation in its curvature It is characterized by an analytical wayof recording certain functions and moreover it is possible tospecify in advance the class of these functions

The sought-after connection is recorded in the form ofparametric equations 119909(119897) and 119910(119897) where parameter 119897 isthe position of a given point along the length of the curveThe determination of 119909(119897) and 119910(119897) needs the expansion ofintegrand functions into Maclaurin series and with regard toinverse curvatures into Taylor series

The determination of coordinates of an adequate tran-sition curve in rectangular coordinate system 119909 119910 makesit possible for an easy transfer of it to Gauss-Kruger (119883 119884)

conformal coordinates and its subsequent layout in fieldmaking use of the satellite measuring technique GNSS

The method can be applied to identify various typesof transition curves The paper gives examples of solvingthe problem involving linear change of curvature and thecurvature in polynomial and trigonometric form

The performed comparative analysis reveals a wide rangeof possibilities which are offered by the presented designmethod Its application enables us to generate diversifiedlayout of the route It simultaneously ensures a completecontrol of correctness of the obtained solutions in view ofsatisfying the geometric and kinematic conditions

The presented method of modelling curvature has auniversal character It can be used for vehicular roads andrailway lines as well In the case of railway lines therearises an additional possibility of modelling in a similar waythe superelevation ramp understood here as a determineddifference in the height of the rail courses

Conflict of Interests

The author declares that he has no conflict of interestsregarding the publication of this paper

References

[1] W Koc and C Specht ldquoApplication of the Polish active GNSSgeodetic network for surveying and design of the railroadrdquo inProceedings of the 1st International Conference on Road and RailInfrastructure (CETRA rsquo10) pp 757ndash762 May 2010

[2] P Lipar M Lakner T Maher andM Zura ldquoEstimation of roadcenterline curvature from raw GPS datardquo Baltic Journal of Roadand Bridge Engineering vol 6 no 3 pp 163ndash168 2011

[3] L Tasci and N Kuloglu ldquoInvestigation of a new transitioncurverdquoTheBaltic Journal of Road and Bridge Engineering vol 6no 1 pp 23ndash29 2011

[4] H-H Cai and G-J Wang ldquoA new method in highway routedesign joining circular arcs by a single C-Bezier curve withshape parameterrdquo Journal of Zhejiang University Science A vol10 no 4 pp 562ndash569 2009

[5] S Dimulyo Z Habib and M Sakai ldquoFair cubic transition bet-ween two circles with one circle inside or tangent to the otherrdquoNumerical Algorithms vol 51 no 4 pp 461ndash476 2009

[6] G Farin ldquoClass A Bezier curvesrdquo Computer Aided GeometricDesign vol 23 no 7 pp 573ndash581 2006

[7] Z Habib and M Sakai ldquoG2 pythagorean hodograph quintictransition between two circles with shape controlrdquo ComputerAided Geometric Design vol 24 no 5 pp 252ndash266 2007

[8] G Harary and A Tal ldquoThe natural 3D spiralrdquo Computer Graph-ics Forum vol 30 no 2 pp 237ndash246 2011

[9] K T Miura ldquoA general equation of aesthetic curves and its self-affinityrdquo Computer-Aided Design and Applications vol 3 no 1ndash4 pp 457ndash464 2006

[10] S H Yahaya M S Salleh and J M Ali ldquoSpur gear design withan S-shaped transition curve application using mathematicaand CAD toolsrdquo in Proceedings of the International Conferenceon Computer Technology and Development (ICCTD rsquo09) pp273ndash276 Dubai United Arab Emirates November 2009

[11] N Yoshida and T Saito ldquoQuasi-aesthetic curves in rationalcubic Bezier formsrdquo Computer-Aided Design and Applicationsvol 4 no 1ndash6 pp 477ndash486 2007

[12] R Ziatdinov N Yoshida and T-W Kim ldquoAnalytic parametricequations of log-aesthetic curves in terms of incomplete gammafunctionsrdquo Computer Aided Geometric Design vol 29 no 2 pp129ndash140 2012

[13] R J Grabowski ldquoSmooth curvilinear transitions along vehicu-lar roads and railway linesrdquo Science Papers of AGH Universityof Science and Technology no 82 Cracow Poland 1984(Polish)

[14] L T Klauder ldquoTrack transition curve geometry based on gegen-bauer polynomialsrdquo in Proceedings of the 6th International Con-ference on Railway Engineering May 2003

[15] W Koc Elements of Design Theory on Track Systems GdanskUniversity of Technology Gdansk Poland 2004 (Polish)

[16] E Mieloszyk and W Koc ldquoGeneral dynamic method for det-ermining transition curve equationsrdquo Rail InternationalmdashSchienen der Welt vol 10 pp 32ndash40 1991

[17] B Kufver ldquoRealigning railway in track renewalsmdashlinear versusS-shaped superelevation rampsrdquo in Proceedings of the 2ndInternational Conference on Railway Engineering May 1999

[18] W Koc and E Mieloszyk ldquoComparative analysis of selectedtransition curves by the use of dynamicmodelrdquoArchives of CivilEngineering vol 33 no 2 pp 239ndash261 1987 (Polish)

[19] S Gleason and D Gebre-Egziabher GNSS Applications andMethods Artech House Publishers Norwood NJ USA 2009

Mathematical Problems in Engineering 13

[20] B Hoffmann-Wellenhof H Lichtenegger and E Wasle GNSSGlobal Navigation Satellite SystemsmdashGPS GLONASS Galileo ampMore Springer Wien Austria 2008

[21] P Misra and P Enge Global Positioning System Signals Mea-surements and Performance Ganga-Jamuna Press LincolnMass USA 2012

[22] C Specht ldquoAccuracy and coverage of the modernized PolishMaritime differential GPS systemrdquo Advances in Space Researchvol 47 no 2 pp 221ndash228 2011

[23] W Koc and K Palikowska ldquoDynamic properties evaluation ofthe selected methods of joining route segments with differentcurvaturerdquo Technika Transportu Szynowego vol 9 pp 1785ndash1807 2012 (Polish)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

ramp The ramp length of course corresponds to the lengthof the transition curveUsing the sameprocedure it is possibleto find the unbalanced acceleration 119886(119897)

3 Transition Curve of Linear Curvature

31 Finding Out the Curvature Equation Linear change ofcurvature along a defined length 119897

119896is obtained by assuming

two principal conditions

119896 (0+) = 1198961

119896 (119897minus

119896) = 1198962

(6)

and a differential equation

11989610158401015840(119897) = 0 (7)

After the determination of constants the solution of thedifferential problem (6) (7) is as follows

119896 (119897) = 1198961+

1

119897119896

(1198962minus 1198961) 119897 (8)

Function Θ(119897) is obtained from formula (5) In the caseunder consideration

Θ (119897) = 1198961119897 +

1

2119897119896

(1198962minus 1198961) 1198972 (9)

32 The Determination of the Transition Curve Length (forRailway Lines) With regard to rail tracks the transition curvemust satisfy two kinematic conditions

120595max le 120595dop (10)

119891max le 119891dop (11)

where 120595max is the maximum value of acceleration incrementon transition curve inms3120595dop is permissible value of accel-eration increment in ms3 119891max is the maximum speed oflifting the wheel on superelevation ramp in mms and 119891dop ispermissible value of speed of lifting the wheel in mms

The unbalanced acceleration 119886(119897) occurring on the tran-sition curve results from speed V of the train magnitude ofcurvature and ordinates of the superelevation ramp itscourse is similar to 119896(119897) and ℎ(119897) Analogously with (8)

119886 (119897) = 1198861+

1

119897119896

(1198862minus 1198861) 119897 (12)

where 1198861and 119886

2are accelerations at ends of the transition

curve

1198861=

V2

rad1198961minus 119892

ℎ1

119904 119886

2=

V2

rad1198962minus 119892

ℎ2

119904

(13)

(ℎ1and ℎ

2are ordinates of the superelevation ramp at its

ends)On the assumption that the speed of the train V = const

the formula for acceleration increment 120595(119897) is as follows

120595 (119897) = V119889

119889119897119886 (119897) =

V119897119896

(1198862minus 1198861) (14)

As can be seen the acceleration increment is here a con-stant value However from condition (10) it follows that

119897120595

119896ge

V120595dop

10038161003816100381610038161198862 minus 1198861

1003816100381610038161003816 (15)

The superelevation ramp equation in the case under con-sideration is similar to (8) and (12)

ℎ (119897) = ℎ1+

1

119897119896

(ℎ2minus ℎ1) 119897 (16)

The speed of lifting the wheel 119891(119897) on the superelevationramp (assuming that the travelling speed of the train V =const) is determined by the equation

119891 (119897) = V119889

119889119897ℎ (119897) =

V119897119896

(ℎ2minus ℎ1) (17)

Thus also119891(119897) is here a constant value but fromcondition(11) it follows that

119897119891

119896ge

V119891dop

1003816100381610038161003816ℎ2 minus ℎ1

1003816100381610038161003816 (18)

The assumed length of the transition curvemust fulfill thecondition

119897119896ge max (119897

120595

119896 119897119891

119896) (19)

33 Coordinates of Transition Curve Connecting Uniform Cur-vatures The determination of 119909(119897) and 119910(119897) by using (4) willneed the expansion of integrands intoMaclaurin series Aftercompleting the whole procedure the following parametricequations are obtained

119909 (119897) = int cosΘ (119897) 119889119897

= 119897 minus1198962

1

61198973minus

1198961

8119897119896

(1198962minus 1198961) 1198974

+ [1198964

1

120minus

1

401198972

119896

(1198962minus 1198961)2] 1198975+

1198963

1

72119897119896

(1198962minus 1198961) 1198976

minus [1198966

1

5040minus

1198962

1

1121198972

119896

(1198962minus 1198961)2] 1198977

minus [1198965

1

1920119897119896

(1198962minus 1198961) minus

1198961

3841198973

119896

(1198962minus 1198961)3] 1198978

+ [1198968

1

362880minus

1198964

1

17281198972

119896

(1198962minus 1198961)2

+1

34561198974

119896

(1198962minus 1198961)4] 1198979sdot sdot sdot

4 Mathematical Problems in Engineering

119910 (119897) = int sinΘ (119897) 119889119897

=1198961

21198972+

1

6119897119896

(1198962minus 1198961) 1198973minus

1198963

1

241198974minus

1198962

1

20119897119896

(1198962minus 1198961) 1198975

+ [1198965

1

720minus

1198961

481198972

119896

(1198962minus 1198961)2] 1198976

+ [1198964

1

336119897119896

(1198962minus 1198961) minus

1

3361198973

119896

(1198962minus 1198961)3] 1198977

minus [1198967

1

40320minus

1198963

1

3841198972

119896

(1198962minus 1198961)2] 1198978

minus [1198966

1

12960119897119896

(1198962minus 1198961) minus

1198962

1

8641198973

119896

(1198962minus 1198961)3] 1198979+ sdot sdot sdot

(20)

Equations (20) for 1198961= 0 describe the curve in the form

of clothoid used to connect a straight with a circular arc

34 Coordinates of Transition Curve Connecting Inverse Cur-vatures At inverse arcs (ie with diversified curvature signs)there arises the problem related to the determined coordi-nates 119909(119897) and 119910(119897) due to the course of function Θ(119897) Forsign 119896

1= sign 119896

2function Θ(119897) is a monotonic function

whereas for sign 1198961

= sign 1198962in the diagram of function Θ(119897)

there appears an extremum at point 1198970 where

Θ1015840(1198970) = 119896 (119897

0) = 1198961+ (1198962minus 1198961)1198970

119897119896

= 0 (21)

Values 1198970and Θ(119897

0) follow from the relations

1198970= minus

1198961

1198962minus 1198961

119897119896

Θ(1198970) = minus

1198962

1

2 (1198962minus 1198961)119897119896

(22)

Figure 2 illustrates a scheme of angle Θ(119897) for the con-nection of inverse circular arcs of radii 119877

1= 1200m and

1198772= 700mThe length of the transition curve 119897

119896= 60m was

determined for speed value V = 90 kmh using the procedureas described at Section 32 (the assumed superelevationvalues were ℎ

1= 20mm and ℎ

2= 45mm)

In this situation the parametric equations of transitioncurve (20) are effective for 119897 isin ⟨0 119897

0⟩ for 119897 isin (119897

0 119897119896⟩ functions

0

0005

001

0015

0 20 40 60 80minus0005

minus001

minus0015

minus002

Q(rad)

l (m)

Figure 2 Diagram of angle Θ(119897) defined by (9) for circulararcs connection of curvatures 119896

1= 11200 radm and 119896

2=

minus1700 radm by means of transition curve of length 119897119896= 60m

cosΘ(119897) and sinΘ(119897) should be expanded to Taylor seriesAfter integration of the equations take the following form

119909 (119897) = int cosΘ (119897) 119889119897

= 119909 (1198970) + cos(minus

1

2

1198962

1

1198962minus 1198961

119897119896) (119897 minus 119897

0)

minus [1

6119897119896

(1198962minus 1198961) sin(minus

1

2

1198962

1

1198962minus 1198961

119897119896)] (119897 minus 119897

0)3

minus [1

401198972

119896

(1198962minus 1198961)2 cos(minus

1

2

1198962

1

1198962minus 1198961

119897119896)] (119897 minus 119897

0)5

+ [1

3361198973

119896

(1198962minus 1198961)3 sin(minus

1

2

1198962

1

1198962minus 1198961

119897119896)] (119897 minus 119897

0)7

+ sdot sdot sdot

(23)

119910 (119897) = int sinΘ (119897) 119889119897

= 119910 (1198970) + sin(minus

1

2

1198962

1

1198962minus 1198961

119897119896) (119897 minus 119897

0)

+ [1

6119897119896

(1198962minus 1198961) cos(minus

1

2

1198962

1

1198962minus 1198961

119897119896)] (119897 minus 119897

0)3

minus [1

401198972

119896

(1198962minus 1198961)2 sin(minus

1

2

1198962

1

1198962minus 1198961

119897119896)] (119897 minus 119897

0)5

minus [1

3361198973

119896

(1198962minus 1198961)3 cos(minus

1

2

1198962

1

1198962minus 1198961

119897119896)] (119897 minus 119897

0)7

+ [1

34561198974

119896

(1198962minus 1198961)4 sin(minus

1

2

1198962

1

1198962minus 1198961

119897119896)] (119897 minus 119897

0)9

+ sdot sdot sdot

(24)

Mathematical Problems in Engineering 5

On completing the parametric equations (20) and (23)along with (24) a correct solution of the task for the caseof diversified curvature signs is obtained (Figure 3) From apractical point of view the determination of the magnitude ofthe tangent slope angle at the end of the transition curve isvery important It amounts to

Θ(119897119896) =

1

2(1198961+ 1198962) 119897119896 (25)

A knowledge of Θ(119897119896) makes it possible to add the tran-

sition curve to the other one having curvature 1198962at its initial

point (satisfying the tangency condition of both the curves)

4 Transition Curve of Polynomial Curvature

From the point of view of vehiclesrsquo dynamics the transitioncurve of linear change of curvature is not the most advan-tageous solution [23] A definitely better one is nonlinearsolution whose characteristics are interesting enough to becomparedwith the Bezier curves preferable lately in literature[4 7]

The differential equation (2) allows to appoint an unlim-ited number of curves with nonlinear curvature change Inthe case of geometric layout of communication routes it willbe appropriate to consider the transition curves of poly-nomial and trigonometric curvature In fact such forms ofcurves are used in the standard ways to connect a straight linewith a circular arc

41 Formulation of the Curvature Equation To formulate thecurvature equation in the polynomial form let us assume thefollowing boundary conditions

119896 (0+) = 1198961 119896 (119897

minus

119896) = 1198962

1198961015840(0+) = 0 119896

1015840(119897minus

119896) = 0

(26)

and the differential equation

119896(4)

(119897) = 0 (27)

After solving the differential problem (26) (27) thefollowing formula for curvature 119896(119897) is obtained

119896 (119897) = 1198961+

3

1198972

119896

(1198962minus 1198961) 1198972minus

2

1198973

119896

(1198962minus 1198961) 1198973 (28)

The form of function Θ(119897) determined on the basis of (5)and (28) is as follows

Θ (119897) = 1198961119897 +

1

1198972

119896

(1198962minus 1198961) 1198973minus

1

21198973

119896

(1198962minus 1198961) 1198974 (29)

On rail routes the length of the transition curve isdetermined like at Section 32 for solution of linear curvatureThe following conditions should be fulfilled here

119897120595

119896ge

3

2

V120595dop

10038161003816100381610038161198862 minus 1198861

1003816100381610038161003816

119897119891

119896ge

3

2

V119891dop

1003816100381610038161003816ℎ2 minus ℎ1

1003816100381610038161003816

(30)

0

01

02

03

04

0 20 40 60 80x (m)

y(m

)

Figure 3 Diagram of horizontal ordinates 119910(119909) defined by (20)(23) and (24) for the joint of circular arcs of curvatures 119896

1=

11200 radm and 1198962

= minus1700 radm with transition curve oflength 119897

119896= 60m (using different horizontal and vertical scales)

The adopted length of the transition curve must satisfycondition (19)

42 Coordinates of Transition Curve Connecting Uniform Cur-vatures On expanding functions cosΘ(119897) and sinΘ(119897) intoMaclaurin series and after integration the following paramet-ric equations are acquired

119909 (119897) = int cosΘ (119897) 119889119897

= 119897 minus1198962

1

61198973+ [

1198964

1

120minus

1198961

51198972

119896

(1198962minus 1198961)] 1198975

+ [1198961

121198973

119896

(1198962minus 1198961)] 1198976

minus [1198966

1

5040minus

1198963

1

421198972

119896

(1198962minus 1198961) +

1

141198974

119896

(1198962minus 1198961)2] 1198977

minus [1198963

1

961198973

119896

(1198962minus 1198961) minus

1

161198975

119896

(1198962minus 1198961)2] 1198978

+ [1198968

1

362880minus

1198965

1

10801198972

119896

(1198962minus 1198961) +

1198962

1

361198974

119896

(1198962minus 1198961)2

+1

721198976

119896

(1198962minus 1198961)2] 1198979+ sdot sdot sdot

119910 (119897) = int sinΘ (119897) 119889119897

=1198961

21198972minus [

1198963

1

24minus

1

41198972

119896

(1198962minus 1198961)] 1198974

minus1

101198973

119896

(1198962minus 1198961) 1198975+ [

1198965

1

720minus

1198962

1

121198972

119896

(1198962minus 1198961)] 1198976

+1198962

1

281198973

119896

(1198962minus 1198961) 1198977

6 Mathematical Problems in Engineering

0

00005

0001

0 20 40 60 80 100l (m)minus00005

minus0001

minus00015

minus0002

k(radm

)

Figure 4 Diagram of curvature 119896(119897) described by (28) for theconnection of circular arcs of curvatures 119896

1= 11200 radm and

1198962

= minus1700 radm with the use of transition curve of length 119897119896

=

90m

minus [1198967

1

40320minus

1198964

1

1921198972

119896

(1198962minus 1198961) +

1198961

161198974

119896

(1198962minus 1198961)2] 1198978

+ [1198961

181198975

119896

(1198962minus 1198961)2minus

1198964

1

4321198973

119896

(1198962minus 1198961)] 1198979+ sdot sdot sdot

(31)

Equations (31) for 1198961

= 0 describe the so-called Blosscurve which can be used to join a straight to a circular arc

43 Coordinates of Transition Curve Connecting Inverse Cur-vatures When the arcs are inverse (ie the curvature signsare different Figure 4) there occurs a problem relating to thedetermined coordinates 119909(119897) and 119910(119897) which results fromfunction Θ(119897) For sign 119896

1= sign 119896

2function Θ(119897) is a mono-

tonic function while for sign 1198961

= sign 1198962in the diagram of

function Θ(119897) there appears extremum at point 1198970 where

Θ1015840(1198970) = 119896 (119897

0) = 1198961+

3

1198972

119896

(1198962minus 1198961) 1198972

0minus

2

1198973

119896

(1198962minus 1198961) 1198973

0= 0

(32)

Parametric equations of transition curve (31) are valid for119897 isin ⟨0 119897

0⟩The determination of value 119897

0makes it necessary to

solve (32) which is cubic The sought-after value 1198970

isin ⟨0 119897119896⟩

satisfying conditions of the problem is defined by the formula

1198970= [

1

2minus cos(

120593

3+

120587

3)] 119897119896 (33)

where angle 120593 follows from the relation

cos120593 =1198961+ 1198962

1198962minus 1198961

(34)

Under such circumstances in order to determine para-metric equations of the transition curve for 119897 isin (119897

0 119897119896⟩ func-

tions cosΘ(119897) and sinΘ(119897) should be expanded into Taylor

series after integration the following parametric equations ofthe transition curve are obtained

119909 (119897) = int cosΘ (119897) 119889119897

= 119909 (1198970) + cosΘ(119897

0) (119897 minus 119897

0)

minus1

6sinΘ(119897

0)Θ10158401015840(1198970) (119897 minus 119897

0)3

minus1

24sinΘ(119897

0)Θ101584010158401015840

(1198970) (119897 minus 119897

0)4

minus 1

40cosΘ(119897

0) [Θ10158401015840(1198970)]2

+1

120sinΘ(119897

0)Θ(4)

(1198970) (119897 minus 119897

0)5

minus1

72cosΘ(119897

0)Θ10158401015840(1198970)Θ101584010158401015840

(1198970) (119897 minus 119897

0)6

+ 1

336sinΘ(119897

0) [Θ10158401015840(1198970)]3

minus1

336cosΘ(119897

0)Θ10158401015840(1198970)Θ(4)

(1198970)

minus1

504cosΘ(119897

0) [Θ101584010158401015840

(1198970)]2

(119897 minus 1198970)7

+ 1

384sinΘ(119897

0) [Θ10158401015840(1198970)]2

Θ101584010158401015840

(1198970)

minus1

1152cosΘ(119897

0)Θ101584010158401015840

(1198970)Θ(4)

(1198970) (119897 minus 119897

0)8

+ 1

3456cosΘ(119897

0) [Θ10158401015840(1198970)]4

+1

1296sinΘ(119897

0)Θ10158401015840(1198970) [Θ101584010158401015840

(1198970)]2

+1

1728sinΘ(119897

0) [Θ10158401015840(1198970)]2

Θ(4)

(1198970)

minus1

10368cosΘ(119897

0) [Θ(4)

(1198970)]2

(119897 minus 1198970)9+ sdot sdot sdot

119910 (119897) = int sinΘ (119897) 119889119897

= 119910 (1198970) + sinΘ(119897

0) (119897 minus 119897

0)

+1

6[cosΘ(119897

0)Θ10158401015840(1198970)] (119897 minus 119897

0)3

+1

24[cosΘ(119897

0)Θ101584010158401015840

(1198970)] (119897 minus 119897

0)4

minus 1

40sinΘ(119897

0) [Θ10158401015840(1198970)]2

minus1

120cosΘ(119897

0)Θ(4)

(1198970) (119897 minus 119897

0)5

minus1

72[sinΘ(119897

0)Θ10158401015840(1198970)Θ101584010158401015840

(1198970)] (119897 minus 119897

0)6

Mathematical Problems in Engineering 7

minus 1

336cosΘ(119897

0) [Θ10158401015840(1198970)]3

+1

336sinΘ(119897

0)Θ10158401015840(1198970)Θ(4)

(1198970)

+1

504sinΘ(119897

0) [Θ101584010158401015840

(1198970)]2

(119897 minus 1198970)7

minus 1

384cosΘ(119897

0) [Θ10158401015840(1198970)]2

Θ101584010158401015840

(1198970)

+1

1152sinΘ(119897

0)Θ101584010158401015840

(1198970)Θ(4)

(1198970) (119897 minus 119897

0)8

+ 1

3456sinΘ(119897

0) [Θ10158401015840(1198970)]4

minus1

1296cosΘ(119897

0)Θ10158401015840(1198970) [Θ101584010158401015840

(1198970)]2

minus1

1728cosΘ(119897

0) [Θ10158401015840(1198970)]2

Θ(4)

(1198970)

minus1

10368sinΘ(119897

0) [Θ(4)

(1198970)]2

(119897 minus 1198970)9+ sdot sdot sdot

(35)

where

Θ(1198970) = 11989611198970+

1

1198972

119896

(1198962minus 1198961) 1198973

0minus

1

21198973

119896

(1198962minus 1198961) 1198974

0

Θ10158401015840(1198970) =

6

1198972

119896

(1198962minus 1198961) 1198970minus

6

1198973

119896

(1198962minus 1198961) 1198972

0

Θ101584010158401015840

(1198970) =

6

1198972

119896

(1198962minus 1198961) minus

12

1198973

119896

(1198962minus 1198961) 1198970

Θ(4)

(1198970) = minus

12

1198973

119896

(1198962minus 1198961)

(36)

Making use of parametric equations (31) and also (35) onecan obtain a correct solution of the problem with diversifiedsigns of curvature

Value of the tangent slope at the end of the transitioncurvemakes it possible to connect to it a curve of curvature 119896

2

satisfying simultaneously the tangency condition of both thecurves The above value is the same as in the case of linearcurvature It is defined by (25)

5 Transition Curve of Curvature inTrigonometric Form

51 Formulation of Curvature Equation As before boundaryconditions (26) and another differential equation areassumed

119896(4)

(119897) +1205872

1198972

119896

11989610158401015840(119897) = 0 (37)

As soon as the differential problem (26) (37) has beensolved a formula for curvature is obtained

119896 (119897) =1

2(1198961+ 1198962) minus

1

2(1198962minus 1198961) cos 120587

119897119896

119897 (38)

The form of function Θ(119897) determined by the use of (5)and (38) is as follows

Θ (119897) =1

2(1198961+ 1198962) 119897 minus

1

2(1198962minus 1198961)119897119896

120587sin 120587

119897119896

119897 (39)

The length of the transition curve on railway routesfollows from conditions

119897120595

119896ge

120587

2

V120595dop

10038161003816100381610038161198862 minus 1198861

1003816100381610038161003816

119897119891

119896ge

120587

2

V119891dop

1003816100381610038161003816ℎ2 minus ℎ1

1003816100381610038161003816

(40)

The assumed length of the curve must satisfy the condi-tion (19)

52 Ordinates of Transition Curve Connecting UniformCurva-tures After the expansion of functions cosΘ(119897) and sinΘ(119897)

intoMaclaurin series to be next integrated the following para-metric equations are obtained

119909 (119897) = int cosΘ (119897) 119889119897

= 119897 minus1198962

1

61198973+ [

1198964

1

120minus

1

60

1205872

1198972

119896

1198961(1198962minus 1198961)] 1198975

minus [1198966

1

5040minus

1

504

1205872

1198972

119896

1198963

1(1198962minus 1198961)

minus1

1680

1205874

1198974

119896

1198961(1198962minus 1198961) minus

1

2016

1205874

1198974

119896

(1198962minus 1198961)2] 1198977

+ [1198968

1

362880minus

1

12960

1205872

1198972

119896

1198965

1(1198962minus 1198961)

minus1

5184

1205874

1198974

119896

1198962

1(1198962minus 1198961)2

minus1

12960

1205874

1198974

119896

1198963

1(1198962minus 1198961) minus

1

90720

1205876

1198976

119896

1198961(1198962minus 1198961)

+1

25920

1205876

1198976

119896

(1198962minus 1198961)2] 1198979+ sdot sdot sdot

8 Mathematical Problems in Engineering

119910 (119897) = int sinΘ (119897) 119889119897

=1

211989611198972minus [

1

241198963

1minus

1

48

1205872

1198972

119896

(1198962minus 1198961)] 1198974

+ [1

7201198965

1minus

1

1441198962

1

1205872

1198972

119896

(1198962minus 1198961)

minus1

1440

1205874

1198974

119896

(1198962minus 1198961)] 1198976

minus [1

403201198967

1minus

1

23041198964

1

1205872

1198972

119896

(1198962minus 1198961)

+1

2304

1205874

1198974

119896

1198961(1198962minus 1198961)2minus

1

3840

1205874

1198974

119896

1198962

1(1198962minus 1198961)

+1

80640

1205876

1198976

119896

(1198962minus 1198961)] 1198978+ sdot sdot sdot

(41)

Equations (41) for 1198961

= 0 describe the curve known ascosinusoid which can be used to connect a straight with acircular arc

53 Transition Curve Coordinates Connecting Inverse Curva-tures Near inverse arcs in the diagram of functionΘ(119897) thereappears the extremum at point 119897

0 where

Θ1015840(1198970) = 119896 (119897

0) =

1

2(1198961+ 1198962) minus

1

2(1198962minus 1198961) cos 120587

119897119896

1198970= 0

(42)

The transition curve parametric equations (41) are effec-tive for 119897 isin ⟨0 119897

0⟩ To determine value 119897

0one must solve (42)

The equation can be written in the form

cos 120587

119897119896

1198970=

1198961+ 1198962

1198962minus 1198961

(43)

It is easy to prove that for sign 1198961

= sign 1198962it is necessary

to satisfy the condition |1198961+1198962| lt |1198962minus1198961| thus |(119896

1+1198962)(1198962minus

1198961)| lt 1 The formula for length 119897

0is as follows

1198970=

1

120587(arccos1198961 + 119896

2

1198962minus 1198961

) 119897119896 (44)

For 119897 isin (1198970 119897119896⟩ functions cosΘ(119897) and sinΘ(119897) should

be expanded into Taylor series The integration provides thefollowing equations

119909 (119897) = int cosΘ (119897) 119889119897

= 119909 (1198970) + cosΘ(119897

0) (119897 minus 119897

0)

minus1

6sinΘ(119897

0)Θ10158401015840(1198970) (119897 minus 119897

0)3

minus1

24sinΘ(119897

0)Θ101584010158401015840

(1198970) sdot (119897 minus 119897

0)4

minus 1

40cosΘ(119897

0) [Θ10158401015840(1198970)]2

+1

120sinΘ(119897

0)Θ(4)

(1198970) (119897 minus 119897

0)5

minus 1

72cosΘ(119897

0)Θ10158401015840(1198970)Θ101584010158401015840

(1198970)

+1

720sinΘ(119897

0)Θ(5)

(1198970) (119897 minus 119897

0)6

+ 1

336sinΘ(119897

0) [Θ10158401015840(1198970)]3

minus1

336cosΘ(119897

0)Θ10158401015840(1198970)Θ(4)

(1198970)

minus1

504cosΘ(119897

0) [Θ101584010158401015840

(1198970)]2

minus1

5040sinΘ(119897

0)Θ(6)

(1198970) (119897 minus 119897

0)7

+ 1

384sinΘ(119897

0) [Θ10158401015840(1198970)]2

Θ101584010158401015840

(1198970)

minus1

1920cosΘ(119897

0)Θ10158401015840(119897) Θ(5)

(119897)

minus1

1152cosΘ(119897

0)Θ101584010158401015840

(1198970)Θ(4)

(1198970)

minus1

40320sinΘ(119897

0)Θ(7)

(1198970) (119897 minus 119897

0)8

+ 1

3456cosΘ(119897

0) [Θ10158401015840(1198970)]4

+1

1296sinΘ(119897

0)Θ10158401015840(1198970) [Θ101584010158401015840

(1198970)]2

+1

1728sinΘ(119897

0) [Θ10158401015840(1198970)]2

Θ(4)

(1198970)

minus1

12960cosΘ(119897

0)Θ10158401015840(1198970)Θ(6)

(1198970)

minus1

6480cosΘ(119897

0)Θ101584010158401015840

(1198970)Θ(5)

(1198970)

minus1

10368cosΘ(119897

0) [Θ(4)

(1198970)]2

minus1

362880sinΘ(119897

0)Θ(8)

(1198970) (119897 minus 119897

0)9+ sdot sdot sdot

Mathematical Problems in Engineering 9

119910 (119897) = int sinΘ (119897) 119889119897

= 119910 (1198970) + sinΘ(119897

0) (119897 minus 119897

0)

+1

6cosΘ(119897

0)Θ10158401015840(1198970) (119897 minus 119897

0)3

+1

24cosΘ(119897

0)Θ101584010158401015840

(1198970) sdot (119897 minus 119897

0)4

minus 1

40sinΘ(119897

0) [Θ10158401015840(1198970)]2

minus1

120cosΘ(119897

0)Θ(4)

(1198970) (119897 minus 119897

0)5

minus 1

72sinΘ(119897

0) 11987610158401015840(1198970)Θ101584010158401015840

(1198970)

minus1

720cosΘ(119897

0)Θ(5)

(1198970) (119897 minus 119897

0)6

minus 1

336cosΘ(119897

0) [Θ10158401015840(1198970)]3

+1

336sinΘ(119897

0)Θ10158401015840(1198970)Θ(4)

(1198970)

+1

504sinΘ(119897

0) [Θ101584010158401015840

(1198970)]2

minus1

5040cosΘ(119897

0)Θ(6)

(1198970) (119897 minus 119897

0)7

minus 1

384cosΘ(119897

0) [Θ10158401015840(1198970)]2

Θ101584010158401015840

(1198970)

+1

1920sinΘ(119897

0)Θ10158401015840(1198970)Θ(5)

(1198970)

+1

1152sinΘ(119897

0)Θ101584010158401015840

(1198970)Θ(4)

(1198970)

minus1

40320cosΘ(119897

0)Θ(7)

(1198970) (119897 minus 119897

0)8

+ 1

3456sinΘ(119897

0) [Θ10158401015840(1198970)]4

minus1

1296cosΘ(119897

0)Θ10158401015840(1198970) [Θ101584010158401015840

(1198970)]2

minus1

1728cosΘ(119897

0) [Θ10158401015840(1198970)]2

Θ(4)

(1198970)

minus1

12960sin119876 (119897

0) 11987610158401015840(1198970) 119876(6)

(1198970)

minus1

6480sin119876 (119897

0) 119876101584010158401015840

(1198970) 119876(5)

(1198970)

minus1

10368sinΘ(119897

0) [Θ(4)

(1198970)]2

+1

362880cosΘ(119897

0)Θ(8)

(1198970) (119897 minus 119897

0)9+ sdot sdot sdot

(45)

where

Θ(1198970) =

1

2(1198961+ 1198962) 1198970plusmn

119897119896

120587radicminus11989611198962

Θ10158401015840(1198970) = ∓

120587

119897119896

radicminus11989611198962

Θ10158401015840(1198970) = ∓

120587

119897119896

radicminus11989611198962

Θ101584010158401015840

(1198970) =

1

2

1205872

1198972

119896

(1198961+ 1198962)

Θ(4)

(1198970) = plusmn

1205873

1198973

119896

radicminus11989611198962

Θ(5)

(1198970) = minus

1

2

1205874

1198974

119896

(1198961+ 1198962)

Θ(6)

(1198970) = ∓

1205875

1198975

119896

radicminus11989611198962

Θ(7)

(1198970) =

1

2

1205876

1198976

119896

(1198961+ 1198962)

Θ(8)

(1198970) = plusmn

1205877

1198977

119896

radicminus11989611198962

(46)

In expressions with double signs plusmn and ∓ the upper onecorresponds to the case 119896

1gt 0 119896

2lt 0 while the lower one

refers to 1198961lt 0 119896

2gt 0

The value of tangent slope at the end of the transitioncurvemakes it possible to connect to it a curve of curvature 119896

2

satisfying the tangency condition of both the curves Thevalue is the same as in the previous cases It is determinedby (25)

6 Comparison of Horizontal Ordinates ofDetermined Transition Curves

61 The Shaping of Horizontal Ordinates of ComparableCurves If the analysis is carried out for railway routes thenthe necessity of meeting kinematic conditions (10) and (11)will require a diversification of the lengths of the comparedwith each other curves (using the same speed of trains)Moreover it is necessary to individually consider cases ofconnecting homogeneous curvatures (sign 119896

1= sign 119896

2) and

inverse arcs (sign 1198961

= sign 1198962)

It is assumed that at joints of consistent arcs with radii1198771= 1200m and 119877

2= 700m an attempt is made to attain a

speed V = 110 kmh Hence it is necessary to employ on arcssuperelevation of ℎ

1= 80mmand ℎ

2= 115mm respectively

The retention of the same kinematic parameters can providediversified lengths of the transition curve

(i) for a curve of linear curvature 119897119896= 40m

(ii) for a curve of polynomial curvature 119897119896= 60m

(iii) for a curve of trigonometric curvature 119897119896= 62 832m

10 Mathematical Problems in Engineering

The above length relations result directly from the equa-tions given at Sections 32 41 and 51

Figure 5 gives diagrams of horizontal ordinates for com-pared transition curves of positive and negative curvaturevalues As seen the curvature ordinates of linear curvatureevidently deviate from the ordinates of other curves In turncurves of polynomial and trigonometric curvature are inprinciple consistent A difference can only be noted in thefinal area And so the class of the function describing thecurvature plays here a significant part with respect to a curveof linear curvature it is a function of class C0 while regardingthe other two curves it is a function of class C1

At the joint of inverse arcs of radii 1198771= 1200m and 119877

2=

700m an attempt is made to reach the speed of V = 90 kmhHence it follows that it is necessary to apply superelevationalong arcs attaining ℎ

1= 20mm and ℎ

2= 45mm respec-

tively By the use the same kinematic parameters it is possibleto obtain the following lengths of the transition curve

(i) for a curve of linear curvature 119897119896= 60m

(ii) for a curve of polynomial curvature 119897119896= 90m

(iii) for a curve of trigonometric curvature 119897119896= 94 248m

Figure 6 presents diagrams of horizontal ordinates ofcompared transition curves for positive and negative valuesof curvature

As can be seen like in the case of consistent arcs the curveordinates of linear curvature deviate significantly from theordinates of the remaining curves but the shape of thesecurves also differs one from the other The class of functiondescribing the curvature is still important However in therange of the same class various geometric solutions arepossible

62 Negotiation of Transition Curves in a Geometric SystemThe transition curves do not appear alone in a geometricsystem but must be connected with some arcs in the neigh-borhood If the input arc is circular then its equation is asfollows

(i) for 1198961gt 0

119910 (119909) = 1198771minus radic1198772

1minus 1199092 (47)

(ii) for 1198961lt 0

119910 (119909) = minus (1198771minus radic1198772

1minus 1199092) (48)

where 119909 lt 0

However if the output arc is also a circular arc then it isexpressed in the following equation

(i) for 1198962gt 0

119910 (119909) = 119910119878minus radic1198772

2minus (119909119878minus 119909)2

119909119878= 119909119870

minus119904119870

radic1 + 1199042

119870

1198772 119910

119878= 119910119870

+1

radic1 + 1199042

119870

1198772

(49)

0051

152

25

0 10 20 30 40 50 60 70x (m)

y(m

)

minus05

minus1

minus15

minus2

minus25

Figure 5 Diagrams of horizontal ordinates of curves joiningconsistent arcs for positive and negative values of curvature (usingdifferent horizontal and vertical scales) curve of linear curvatureline in violet colour curve of polynomial curvature in green colourand curve of trigonometric curvature in red colour

0

05

1

15

0 20 40 60 80 100x (m)

y(m

)

minus05

minus1

minus15

Figure 6 Diagrams of horizontal ordinates of curves connectinginverse arcs for positive and negative values of curvature (usingdifferent horizontal and vertical scales) the curve of linear curvatureis in violet color the curve of polynomial curvature is in green colorand the curve of trigonometric curvature is in red color

(ii) for 1198962lt 0

119910 (119909) = 119910119878+ radic1198772

2minus (119909119878minus 119909)2

119909119878= 119909119870

+119904119870

radic1 + 1199042

119870

1198772 119910

119878= 119910119870

minus1

radic1 + 1199042

119870

1198772

(50)

where 119909 gt 119909119870

In (49) and (50) 119909119870denotes the abscissae of the transition

curve end119910119870is the ordinate of the transition curve endwhile

119904119870is the value of tangent at the end of the transition curve

Values for numerical examples under consideration are givenin Table 1

Figures 7 and 8 illustrate the geometric systems made upof two circular arcs of consistent curvature connected withvarious types of transition curves As seen the application of agiven type of transition curve leads to an adequate position ofthe output circular arcThe position of this arc by the use of alinear transition curve clearly deviates from such cases whereadvantage is taken of curves with polynomial and trigono-metrical curvatureWith regard to the latter amutual positionof the circular arc varies insignificantly

Mathematical Problems in Engineering 11

Table 1 Comparison of values 119909119870 119910119870 and 119904

119870for numerical examples analyzed

Curve 119897119896[m] Θ(119897

119896) [rad] 119904

119870119909119870[m] 119910

119870[m]

Case 1 1198961= 11200 radm and 119896

2= 1700 radm

Curve C0 40 0045238 0045269 39988 0825Curve C1 (p) 60 0067857 0067961 59963 1821Curve C1 (t) 62832 0071060 0071180 62796 1940

Case 2 1198961= 11200 radm and 119896

2= minus1700 radm

Curve C0 60 minus0017857 minus0017859 59999 0143Curve C1 (p) 90 minus0026786 minus0026792 89991 0627Curve C1 (t) 94248 minus0028050 minus0028057 94238 0714

Case 3 1198961= minus11200 radm and 119896

2= 1700 radm

Curve C0 60 001786 0017859 59999 minus0143Curve C1 (p) 90 0026786 0026792 89991 minus0627Curve C1 (t) 94248 0028050 0028057 94238 minus0714

Case 4 1198961= minus11200 radm and 119896

2= minus1700 radm

Curve C0 40 minus0045238 minus0045269 39988 minus0825Curve C1 (p) 60 minus0067857 minus0067961 59969 minus1821Curve C1 (t) 62832 minus0071060 minus0071180 62976 minus1940

0

1

2

3

4

5

6

0 20 40 60 80 100 120x (m)

y(m

)

minus40 minus20

Figure 7 Examples of geometric systems consisting of two circulararcs with consistent curvature (brown line) connected with differenttransition curves for positive values of curvature (using differenthorizontal and vertical scales) curve of linear curvature in violetline curve of polynomial curvature in green and curve of trigono-metric curvature in red

00 20 40 60 80 100 120

x (m)

y(m

)

minus40 minus20

minus1

minus2

minus3

minus4

minus5

minus6

minus7

Figure 8 Examples of geometric systems consisting of two circulararcs with consistent curvature (brown line) connected with differenttransition curves for negative values of curvature (using differenthorizontal and vertical scales) curve of linear curvature in violetline curve of polynomial curvature in green and curve of trigono-metric curvature in red

0

05

1

15

0 50 100 150x (m)

y(m

)

minus05

minus1

minus15

minus2

minus50

Figure 9 Examples of geometric systems consisting of two inversecircular arcs (brown line) connected with different types of transi-tion curves for positive values of input curvature (using differenthorizontal and vertical scales) curve of linear curvature in violetline curve of polynomial curvature in green and curve of trigono-metric curvature in red

Figures 9 and 10 illustrate geometric systems made up oftwo inverse circular arcs connected with various types oftransition curves

Here the layout of the output circular arc is much morediversifiedThe position of the arc obtained by the applicationof the transition curve of linear curvature diverts by severaldozenmeters from situationswhere curves of polynomial andtrigonometric curvature have been used The difference forcurves of polynomial and trigonometric curvature attains inthe numerical example analyzed an order of several meters

The presented examples are taken in the very nature ofthings at random They reveal wide possibilities for the pre-sented analytical method in modelling geometric systems Itsapplication makes it possible to generate diversified layout ofthe communication routes adapted to the field conditions andother circumstances Simultaneously the method ensurescorrectness of the obtained solution in view of the kinematicparameters magnitudes in force

12 Mathematical Problems in Engineering

0

05

1

15

2

0 50 100 150x (m)

y(m

)

minus05

minus1

minus15

minus50

Figure 10 Examples of geometric systems consisting of twoinverse circular arcs (brown line) connected with different typesof transition curves for negative values of input curvature (usingdifferent horizontal and vertical scales) curve of linear curvaturein violet line curve of polynomial curvature in green and curve oftrigonometric curvature in red

7 Conclusions

The presented universal method of modelling geometricalsystem of a communication route is based on the determina-tion of an adequate curvature equation by using differentialequationsThe technique enables us to design connections ofvarious geometrical elements of the route indicating diversifi-cation in its curvature It is characterized by an analytical wayof recording certain functions and moreover it is possible tospecify in advance the class of these functions

The sought-after connection is recorded in the form ofparametric equations 119909(119897) and 119910(119897) where parameter 119897 isthe position of a given point along the length of the curveThe determination of 119909(119897) and 119910(119897) needs the expansion ofintegrand functions into Maclaurin series and with regard toinverse curvatures into Taylor series

The determination of coordinates of an adequate tran-sition curve in rectangular coordinate system 119909 119910 makesit possible for an easy transfer of it to Gauss-Kruger (119883 119884)

conformal coordinates and its subsequent layout in fieldmaking use of the satellite measuring technique GNSS

The method can be applied to identify various typesof transition curves The paper gives examples of solvingthe problem involving linear change of curvature and thecurvature in polynomial and trigonometric form

The performed comparative analysis reveals a wide rangeof possibilities which are offered by the presented designmethod Its application enables us to generate diversifiedlayout of the route It simultaneously ensures a completecontrol of correctness of the obtained solutions in view ofsatisfying the geometric and kinematic conditions

The presented method of modelling curvature has auniversal character It can be used for vehicular roads andrailway lines as well In the case of railway lines therearises an additional possibility of modelling in a similar waythe superelevation ramp understood here as a determineddifference in the height of the rail courses

Conflict of Interests

The author declares that he has no conflict of interestsregarding the publication of this paper

References

[1] W Koc and C Specht ldquoApplication of the Polish active GNSSgeodetic network for surveying and design of the railroadrdquo inProceedings of the 1st International Conference on Road and RailInfrastructure (CETRA rsquo10) pp 757ndash762 May 2010

[2] P Lipar M Lakner T Maher andM Zura ldquoEstimation of roadcenterline curvature from raw GPS datardquo Baltic Journal of Roadand Bridge Engineering vol 6 no 3 pp 163ndash168 2011

[3] L Tasci and N Kuloglu ldquoInvestigation of a new transitioncurverdquoTheBaltic Journal of Road and Bridge Engineering vol 6no 1 pp 23ndash29 2011

[4] H-H Cai and G-J Wang ldquoA new method in highway routedesign joining circular arcs by a single C-Bezier curve withshape parameterrdquo Journal of Zhejiang University Science A vol10 no 4 pp 562ndash569 2009

[5] S Dimulyo Z Habib and M Sakai ldquoFair cubic transition bet-ween two circles with one circle inside or tangent to the otherrdquoNumerical Algorithms vol 51 no 4 pp 461ndash476 2009

[6] G Farin ldquoClass A Bezier curvesrdquo Computer Aided GeometricDesign vol 23 no 7 pp 573ndash581 2006

[7] Z Habib and M Sakai ldquoG2 pythagorean hodograph quintictransition between two circles with shape controlrdquo ComputerAided Geometric Design vol 24 no 5 pp 252ndash266 2007

[8] G Harary and A Tal ldquoThe natural 3D spiralrdquo Computer Graph-ics Forum vol 30 no 2 pp 237ndash246 2011

[9] K T Miura ldquoA general equation of aesthetic curves and its self-affinityrdquo Computer-Aided Design and Applications vol 3 no 1ndash4 pp 457ndash464 2006

[10] S H Yahaya M S Salleh and J M Ali ldquoSpur gear design withan S-shaped transition curve application using mathematicaand CAD toolsrdquo in Proceedings of the International Conferenceon Computer Technology and Development (ICCTD rsquo09) pp273ndash276 Dubai United Arab Emirates November 2009

[11] N Yoshida and T Saito ldquoQuasi-aesthetic curves in rationalcubic Bezier formsrdquo Computer-Aided Design and Applicationsvol 4 no 1ndash6 pp 477ndash486 2007

[12] R Ziatdinov N Yoshida and T-W Kim ldquoAnalytic parametricequations of log-aesthetic curves in terms of incomplete gammafunctionsrdquo Computer Aided Geometric Design vol 29 no 2 pp129ndash140 2012

[13] R J Grabowski ldquoSmooth curvilinear transitions along vehicu-lar roads and railway linesrdquo Science Papers of AGH Universityof Science and Technology no 82 Cracow Poland 1984(Polish)

[14] L T Klauder ldquoTrack transition curve geometry based on gegen-bauer polynomialsrdquo in Proceedings of the 6th International Con-ference on Railway Engineering May 2003

[15] W Koc Elements of Design Theory on Track Systems GdanskUniversity of Technology Gdansk Poland 2004 (Polish)

[16] E Mieloszyk and W Koc ldquoGeneral dynamic method for det-ermining transition curve equationsrdquo Rail InternationalmdashSchienen der Welt vol 10 pp 32ndash40 1991

[17] B Kufver ldquoRealigning railway in track renewalsmdashlinear versusS-shaped superelevation rampsrdquo in Proceedings of the 2ndInternational Conference on Railway Engineering May 1999

[18] W Koc and E Mieloszyk ldquoComparative analysis of selectedtransition curves by the use of dynamicmodelrdquoArchives of CivilEngineering vol 33 no 2 pp 239ndash261 1987 (Polish)

[19] S Gleason and D Gebre-Egziabher GNSS Applications andMethods Artech House Publishers Norwood NJ USA 2009

Mathematical Problems in Engineering 13

[20] B Hoffmann-Wellenhof H Lichtenegger and E Wasle GNSSGlobal Navigation Satellite SystemsmdashGPS GLONASS Galileo ampMore Springer Wien Austria 2008

[21] P Misra and P Enge Global Positioning System Signals Mea-surements and Performance Ganga-Jamuna Press LincolnMass USA 2012

[22] C Specht ldquoAccuracy and coverage of the modernized PolishMaritime differential GPS systemrdquo Advances in Space Researchvol 47 no 2 pp 221ndash228 2011

[23] W Koc and K Palikowska ldquoDynamic properties evaluation ofthe selected methods of joining route segments with differentcurvaturerdquo Technika Transportu Szynowego vol 9 pp 1785ndash1807 2012 (Polish)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

119910 (119897) = int sinΘ (119897) 119889119897

=1198961

21198972+

1

6119897119896

(1198962minus 1198961) 1198973minus

1198963

1

241198974minus

1198962

1

20119897119896

(1198962minus 1198961) 1198975

+ [1198965

1

720minus

1198961

481198972

119896

(1198962minus 1198961)2] 1198976

+ [1198964

1

336119897119896

(1198962minus 1198961) minus

1

3361198973

119896

(1198962minus 1198961)3] 1198977

minus [1198967

1

40320minus

1198963

1

3841198972

119896

(1198962minus 1198961)2] 1198978

minus [1198966

1

12960119897119896

(1198962minus 1198961) minus

1198962

1

8641198973

119896

(1198962minus 1198961)3] 1198979+ sdot sdot sdot

(20)

Equations (20) for 1198961= 0 describe the curve in the form

of clothoid used to connect a straight with a circular arc

34 Coordinates of Transition Curve Connecting Inverse Cur-vatures At inverse arcs (ie with diversified curvature signs)there arises the problem related to the determined coordi-nates 119909(119897) and 119910(119897) due to the course of function Θ(119897) Forsign 119896

1= sign 119896

2function Θ(119897) is a monotonic function

whereas for sign 1198961

= sign 1198962in the diagram of function Θ(119897)

there appears an extremum at point 1198970 where

Θ1015840(1198970) = 119896 (119897

0) = 1198961+ (1198962minus 1198961)1198970

119897119896

= 0 (21)

Values 1198970and Θ(119897

0) follow from the relations

1198970= minus

1198961

1198962minus 1198961

119897119896

Θ(1198970) = minus

1198962

1

2 (1198962minus 1198961)119897119896

(22)

Figure 2 illustrates a scheme of angle Θ(119897) for the con-nection of inverse circular arcs of radii 119877

1= 1200m and

1198772= 700mThe length of the transition curve 119897

119896= 60m was

determined for speed value V = 90 kmh using the procedureas described at Section 32 (the assumed superelevationvalues were ℎ

1= 20mm and ℎ

2= 45mm)

In this situation the parametric equations of transitioncurve (20) are effective for 119897 isin ⟨0 119897

0⟩ for 119897 isin (119897

0 119897119896⟩ functions

0

0005

001

0015

0 20 40 60 80minus0005

minus001

minus0015

minus002

Q(rad)

l (m)

Figure 2 Diagram of angle Θ(119897) defined by (9) for circulararcs connection of curvatures 119896

1= 11200 radm and 119896

2=

minus1700 radm by means of transition curve of length 119897119896= 60m

cosΘ(119897) and sinΘ(119897) should be expanded to Taylor seriesAfter integration of the equations take the following form

119909 (119897) = int cosΘ (119897) 119889119897

= 119909 (1198970) + cos(minus

1

2

1198962

1

1198962minus 1198961

119897119896) (119897 minus 119897

0)

minus [1

6119897119896

(1198962minus 1198961) sin(minus

1

2

1198962

1

1198962minus 1198961

119897119896)] (119897 minus 119897

0)3

minus [1

401198972

119896

(1198962minus 1198961)2 cos(minus

1

2

1198962

1

1198962minus 1198961

119897119896)] (119897 minus 119897

0)5

+ [1

3361198973

119896

(1198962minus 1198961)3 sin(minus

1

2

1198962

1

1198962minus 1198961

119897119896)] (119897 minus 119897

0)7

+ sdot sdot sdot

(23)

119910 (119897) = int sinΘ (119897) 119889119897

= 119910 (1198970) + sin(minus

1

2

1198962

1

1198962minus 1198961

119897119896) (119897 minus 119897

0)

+ [1

6119897119896

(1198962minus 1198961) cos(minus

1

2

1198962

1

1198962minus 1198961

119897119896)] (119897 minus 119897

0)3

minus [1

401198972

119896

(1198962minus 1198961)2 sin(minus

1

2

1198962

1

1198962minus 1198961

119897119896)] (119897 minus 119897

0)5

minus [1

3361198973

119896

(1198962minus 1198961)3 cos(minus

1

2

1198962

1

1198962minus 1198961

119897119896)] (119897 minus 119897

0)7

+ [1

34561198974

119896

(1198962minus 1198961)4 sin(minus

1

2

1198962

1

1198962minus 1198961

119897119896)] (119897 minus 119897

0)9

+ sdot sdot sdot

(24)

Mathematical Problems in Engineering 5

On completing the parametric equations (20) and (23)along with (24) a correct solution of the task for the caseof diversified curvature signs is obtained (Figure 3) From apractical point of view the determination of the magnitude ofthe tangent slope angle at the end of the transition curve isvery important It amounts to

Θ(119897119896) =

1

2(1198961+ 1198962) 119897119896 (25)

A knowledge of Θ(119897119896) makes it possible to add the tran-

sition curve to the other one having curvature 1198962at its initial

point (satisfying the tangency condition of both the curves)

4 Transition Curve of Polynomial Curvature

From the point of view of vehiclesrsquo dynamics the transitioncurve of linear change of curvature is not the most advan-tageous solution [23] A definitely better one is nonlinearsolution whose characteristics are interesting enough to becomparedwith the Bezier curves preferable lately in literature[4 7]

The differential equation (2) allows to appoint an unlim-ited number of curves with nonlinear curvature change Inthe case of geometric layout of communication routes it willbe appropriate to consider the transition curves of poly-nomial and trigonometric curvature In fact such forms ofcurves are used in the standard ways to connect a straight linewith a circular arc

41 Formulation of the Curvature Equation To formulate thecurvature equation in the polynomial form let us assume thefollowing boundary conditions

119896 (0+) = 1198961 119896 (119897

minus

119896) = 1198962

1198961015840(0+) = 0 119896

1015840(119897minus

119896) = 0

(26)

and the differential equation

119896(4)

(119897) = 0 (27)

After solving the differential problem (26) (27) thefollowing formula for curvature 119896(119897) is obtained

119896 (119897) = 1198961+

3

1198972

119896

(1198962minus 1198961) 1198972minus

2

1198973

119896

(1198962minus 1198961) 1198973 (28)

The form of function Θ(119897) determined on the basis of (5)and (28) is as follows

Θ (119897) = 1198961119897 +

1

1198972

119896

(1198962minus 1198961) 1198973minus

1

21198973

119896

(1198962minus 1198961) 1198974 (29)

On rail routes the length of the transition curve isdetermined like at Section 32 for solution of linear curvatureThe following conditions should be fulfilled here

119897120595

119896ge

3

2

V120595dop

10038161003816100381610038161198862 minus 1198861

1003816100381610038161003816

119897119891

119896ge

3

2

V119891dop

1003816100381610038161003816ℎ2 minus ℎ1

1003816100381610038161003816

(30)

0

01

02

03

04

0 20 40 60 80x (m)

y(m

)

Figure 3 Diagram of horizontal ordinates 119910(119909) defined by (20)(23) and (24) for the joint of circular arcs of curvatures 119896

1=

11200 radm and 1198962

= minus1700 radm with transition curve oflength 119897

119896= 60m (using different horizontal and vertical scales)

The adopted length of the transition curve must satisfycondition (19)

42 Coordinates of Transition Curve Connecting Uniform Cur-vatures On expanding functions cosΘ(119897) and sinΘ(119897) intoMaclaurin series and after integration the following paramet-ric equations are acquired

119909 (119897) = int cosΘ (119897) 119889119897

= 119897 minus1198962

1

61198973+ [

1198964

1

120minus

1198961

51198972

119896

(1198962minus 1198961)] 1198975

+ [1198961

121198973

119896

(1198962minus 1198961)] 1198976

minus [1198966

1

5040minus

1198963

1

421198972

119896

(1198962minus 1198961) +

1

141198974

119896

(1198962minus 1198961)2] 1198977

minus [1198963

1

961198973

119896

(1198962minus 1198961) minus

1

161198975

119896

(1198962minus 1198961)2] 1198978

+ [1198968

1

362880minus

1198965

1

10801198972

119896

(1198962minus 1198961) +

1198962

1

361198974

119896

(1198962minus 1198961)2

+1

721198976

119896

(1198962minus 1198961)2] 1198979+ sdot sdot sdot

119910 (119897) = int sinΘ (119897) 119889119897

=1198961

21198972minus [

1198963

1

24minus

1

41198972

119896

(1198962minus 1198961)] 1198974

minus1

101198973

119896

(1198962minus 1198961) 1198975+ [

1198965

1

720minus

1198962

1

121198972

119896

(1198962minus 1198961)] 1198976

+1198962

1

281198973

119896

(1198962minus 1198961) 1198977

6 Mathematical Problems in Engineering

0

00005

0001

0 20 40 60 80 100l (m)minus00005

minus0001

minus00015

minus0002

k(radm

)

Figure 4 Diagram of curvature 119896(119897) described by (28) for theconnection of circular arcs of curvatures 119896

1= 11200 radm and

1198962

= minus1700 radm with the use of transition curve of length 119897119896

=

90m

minus [1198967

1

40320minus

1198964

1

1921198972

119896

(1198962minus 1198961) +

1198961

161198974

119896

(1198962minus 1198961)2] 1198978

+ [1198961

181198975

119896

(1198962minus 1198961)2minus

1198964

1

4321198973

119896

(1198962minus 1198961)] 1198979+ sdot sdot sdot

(31)

Equations (31) for 1198961

= 0 describe the so-called Blosscurve which can be used to join a straight to a circular arc

43 Coordinates of Transition Curve Connecting Inverse Cur-vatures When the arcs are inverse (ie the curvature signsare different Figure 4) there occurs a problem relating to thedetermined coordinates 119909(119897) and 119910(119897) which results fromfunction Θ(119897) For sign 119896

1= sign 119896

2function Θ(119897) is a mono-

tonic function while for sign 1198961

= sign 1198962in the diagram of

function Θ(119897) there appears extremum at point 1198970 where

Θ1015840(1198970) = 119896 (119897

0) = 1198961+

3

1198972

119896

(1198962minus 1198961) 1198972

0minus

2

1198973

119896

(1198962minus 1198961) 1198973

0= 0

(32)

Parametric equations of transition curve (31) are valid for119897 isin ⟨0 119897

0⟩The determination of value 119897

0makes it necessary to

solve (32) which is cubic The sought-after value 1198970

isin ⟨0 119897119896⟩

satisfying conditions of the problem is defined by the formula

1198970= [

1

2minus cos(

120593

3+

120587

3)] 119897119896 (33)

where angle 120593 follows from the relation

cos120593 =1198961+ 1198962

1198962minus 1198961

(34)

Under such circumstances in order to determine para-metric equations of the transition curve for 119897 isin (119897

0 119897119896⟩ func-

tions cosΘ(119897) and sinΘ(119897) should be expanded into Taylor

series after integration the following parametric equations ofthe transition curve are obtained

119909 (119897) = int cosΘ (119897) 119889119897

= 119909 (1198970) + cosΘ(119897

0) (119897 minus 119897

0)

minus1

6sinΘ(119897

0)Θ10158401015840(1198970) (119897 minus 119897

0)3

minus1

24sinΘ(119897

0)Θ101584010158401015840

(1198970) (119897 minus 119897

0)4

minus 1

40cosΘ(119897

0) [Θ10158401015840(1198970)]2

+1

120sinΘ(119897

0)Θ(4)

(1198970) (119897 minus 119897

0)5

minus1

72cosΘ(119897

0)Θ10158401015840(1198970)Θ101584010158401015840

(1198970) (119897 minus 119897

0)6

+ 1

336sinΘ(119897

0) [Θ10158401015840(1198970)]3

minus1

336cosΘ(119897

0)Θ10158401015840(1198970)Θ(4)

(1198970)

minus1

504cosΘ(119897

0) [Θ101584010158401015840

(1198970)]2

(119897 minus 1198970)7

+ 1

384sinΘ(119897

0) [Θ10158401015840(1198970)]2

Θ101584010158401015840

(1198970)

minus1

1152cosΘ(119897

0)Θ101584010158401015840

(1198970)Θ(4)

(1198970) (119897 minus 119897

0)8

+ 1

3456cosΘ(119897

0) [Θ10158401015840(1198970)]4

+1

1296sinΘ(119897

0)Θ10158401015840(1198970) [Θ101584010158401015840

(1198970)]2

+1

1728sinΘ(119897

0) [Θ10158401015840(1198970)]2

Θ(4)

(1198970)

minus1

10368cosΘ(119897

0) [Θ(4)

(1198970)]2

(119897 minus 1198970)9+ sdot sdot sdot

119910 (119897) = int sinΘ (119897) 119889119897

= 119910 (1198970) + sinΘ(119897

0) (119897 minus 119897

0)

+1

6[cosΘ(119897

0)Θ10158401015840(1198970)] (119897 minus 119897

0)3

+1

24[cosΘ(119897

0)Θ101584010158401015840

(1198970)] (119897 minus 119897

0)4

minus 1

40sinΘ(119897

0) [Θ10158401015840(1198970)]2

minus1

120cosΘ(119897

0)Θ(4)

(1198970) (119897 minus 119897

0)5

minus1

72[sinΘ(119897

0)Θ10158401015840(1198970)Θ101584010158401015840

(1198970)] (119897 minus 119897

0)6

Mathematical Problems in Engineering 7

minus 1

336cosΘ(119897

0) [Θ10158401015840(1198970)]3

+1

336sinΘ(119897

0)Θ10158401015840(1198970)Θ(4)

(1198970)

+1

504sinΘ(119897

0) [Θ101584010158401015840

(1198970)]2

(119897 minus 1198970)7

minus 1

384cosΘ(119897

0) [Θ10158401015840(1198970)]2

Θ101584010158401015840

(1198970)

+1

1152sinΘ(119897

0)Θ101584010158401015840

(1198970)Θ(4)

(1198970) (119897 minus 119897

0)8

+ 1

3456sinΘ(119897

0) [Θ10158401015840(1198970)]4

minus1

1296cosΘ(119897

0)Θ10158401015840(1198970) [Θ101584010158401015840

(1198970)]2

minus1

1728cosΘ(119897

0) [Θ10158401015840(1198970)]2

Θ(4)

(1198970)

minus1

10368sinΘ(119897

0) [Θ(4)

(1198970)]2

(119897 minus 1198970)9+ sdot sdot sdot

(35)

where

Θ(1198970) = 11989611198970+

1

1198972

119896

(1198962minus 1198961) 1198973

0minus

1

21198973

119896

(1198962minus 1198961) 1198974

0

Θ10158401015840(1198970) =

6

1198972

119896

(1198962minus 1198961) 1198970minus

6

1198973

119896

(1198962minus 1198961) 1198972

0

Θ101584010158401015840

(1198970) =

6

1198972

119896

(1198962minus 1198961) minus

12

1198973

119896

(1198962minus 1198961) 1198970

Θ(4)

(1198970) = minus

12

1198973

119896

(1198962minus 1198961)

(36)

Making use of parametric equations (31) and also (35) onecan obtain a correct solution of the problem with diversifiedsigns of curvature

Value of the tangent slope at the end of the transitioncurvemakes it possible to connect to it a curve of curvature 119896

2

satisfying simultaneously the tangency condition of both thecurves The above value is the same as in the case of linearcurvature It is defined by (25)

5 Transition Curve of Curvature inTrigonometric Form

51 Formulation of Curvature Equation As before boundaryconditions (26) and another differential equation areassumed

119896(4)

(119897) +1205872

1198972

119896

11989610158401015840(119897) = 0 (37)

As soon as the differential problem (26) (37) has beensolved a formula for curvature is obtained

119896 (119897) =1

2(1198961+ 1198962) minus

1

2(1198962minus 1198961) cos 120587

119897119896

119897 (38)

The form of function Θ(119897) determined by the use of (5)and (38) is as follows

Θ (119897) =1

2(1198961+ 1198962) 119897 minus

1

2(1198962minus 1198961)119897119896

120587sin 120587

119897119896

119897 (39)

The length of the transition curve on railway routesfollows from conditions

119897120595

119896ge

120587

2

V120595dop

10038161003816100381610038161198862 minus 1198861

1003816100381610038161003816

119897119891

119896ge

120587

2

V119891dop

1003816100381610038161003816ℎ2 minus ℎ1

1003816100381610038161003816

(40)

The assumed length of the curve must satisfy the condi-tion (19)

52 Ordinates of Transition Curve Connecting UniformCurva-tures After the expansion of functions cosΘ(119897) and sinΘ(119897)

intoMaclaurin series to be next integrated the following para-metric equations are obtained

119909 (119897) = int cosΘ (119897) 119889119897

= 119897 minus1198962

1

61198973+ [

1198964

1

120minus

1

60

1205872

1198972

119896

1198961(1198962minus 1198961)] 1198975

minus [1198966

1

5040minus

1

504

1205872

1198972

119896

1198963

1(1198962minus 1198961)

minus1

1680

1205874

1198974

119896

1198961(1198962minus 1198961) minus

1

2016

1205874

1198974

119896

(1198962minus 1198961)2] 1198977

+ [1198968

1

362880minus

1

12960

1205872

1198972

119896

1198965

1(1198962minus 1198961)

minus1

5184

1205874

1198974

119896

1198962

1(1198962minus 1198961)2

minus1

12960

1205874

1198974

119896

1198963

1(1198962minus 1198961) minus

1

90720

1205876

1198976

119896

1198961(1198962minus 1198961)

+1

25920

1205876

1198976

119896

(1198962minus 1198961)2] 1198979+ sdot sdot sdot

8 Mathematical Problems in Engineering

119910 (119897) = int sinΘ (119897) 119889119897

=1

211989611198972minus [

1

241198963

1minus

1

48

1205872

1198972

119896

(1198962minus 1198961)] 1198974

+ [1

7201198965

1minus

1

1441198962

1

1205872

1198972

119896

(1198962minus 1198961)

minus1

1440

1205874

1198974

119896

(1198962minus 1198961)] 1198976

minus [1

403201198967

1minus

1

23041198964

1

1205872

1198972

119896

(1198962minus 1198961)

+1

2304

1205874

1198974

119896

1198961(1198962minus 1198961)2minus

1

3840

1205874

1198974

119896

1198962

1(1198962minus 1198961)

+1

80640

1205876

1198976

119896

(1198962minus 1198961)] 1198978+ sdot sdot sdot

(41)

Equations (41) for 1198961

= 0 describe the curve known ascosinusoid which can be used to connect a straight with acircular arc

53 Transition Curve Coordinates Connecting Inverse Curva-tures Near inverse arcs in the diagram of functionΘ(119897) thereappears the extremum at point 119897

0 where

Θ1015840(1198970) = 119896 (119897

0) =

1

2(1198961+ 1198962) minus

1

2(1198962minus 1198961) cos 120587

119897119896

1198970= 0

(42)

The transition curve parametric equations (41) are effec-tive for 119897 isin ⟨0 119897

0⟩ To determine value 119897

0one must solve (42)

The equation can be written in the form

cos 120587

119897119896

1198970=

1198961+ 1198962

1198962minus 1198961

(43)

It is easy to prove that for sign 1198961

= sign 1198962it is necessary

to satisfy the condition |1198961+1198962| lt |1198962minus1198961| thus |(119896

1+1198962)(1198962minus

1198961)| lt 1 The formula for length 119897

0is as follows

1198970=

1

120587(arccos1198961 + 119896

2

1198962minus 1198961

) 119897119896 (44)

For 119897 isin (1198970 119897119896⟩ functions cosΘ(119897) and sinΘ(119897) should

be expanded into Taylor series The integration provides thefollowing equations

119909 (119897) = int cosΘ (119897) 119889119897

= 119909 (1198970) + cosΘ(119897

0) (119897 minus 119897

0)

minus1

6sinΘ(119897

0)Θ10158401015840(1198970) (119897 minus 119897

0)3

minus1

24sinΘ(119897

0)Θ101584010158401015840

(1198970) sdot (119897 minus 119897

0)4

minus 1

40cosΘ(119897

0) [Θ10158401015840(1198970)]2

+1

120sinΘ(119897

0)Θ(4)

(1198970) (119897 minus 119897

0)5

minus 1

72cosΘ(119897

0)Θ10158401015840(1198970)Θ101584010158401015840

(1198970)

+1

720sinΘ(119897

0)Θ(5)

(1198970) (119897 minus 119897

0)6

+ 1

336sinΘ(119897

0) [Θ10158401015840(1198970)]3

minus1

336cosΘ(119897

0)Θ10158401015840(1198970)Θ(4)

(1198970)

minus1

504cosΘ(119897

0) [Θ101584010158401015840

(1198970)]2

minus1

5040sinΘ(119897

0)Θ(6)

(1198970) (119897 minus 119897

0)7

+ 1

384sinΘ(119897

0) [Θ10158401015840(1198970)]2

Θ101584010158401015840

(1198970)

minus1

1920cosΘ(119897

0)Θ10158401015840(119897) Θ(5)

(119897)

minus1

1152cosΘ(119897

0)Θ101584010158401015840

(1198970)Θ(4)

(1198970)

minus1

40320sinΘ(119897

0)Θ(7)

(1198970) (119897 minus 119897

0)8

+ 1

3456cosΘ(119897

0) [Θ10158401015840(1198970)]4

+1

1296sinΘ(119897

0)Θ10158401015840(1198970) [Θ101584010158401015840

(1198970)]2

+1

1728sinΘ(119897

0) [Θ10158401015840(1198970)]2

Θ(4)

(1198970)

minus1

12960cosΘ(119897

0)Θ10158401015840(1198970)Θ(6)

(1198970)

minus1

6480cosΘ(119897

0)Θ101584010158401015840

(1198970)Θ(5)

(1198970)

minus1

10368cosΘ(119897

0) [Θ(4)

(1198970)]2

minus1

362880sinΘ(119897

0)Θ(8)

(1198970) (119897 minus 119897

0)9+ sdot sdot sdot

Mathematical Problems in Engineering 9

119910 (119897) = int sinΘ (119897) 119889119897

= 119910 (1198970) + sinΘ(119897

0) (119897 minus 119897

0)

+1

6cosΘ(119897

0)Θ10158401015840(1198970) (119897 minus 119897

0)3

+1

24cosΘ(119897

0)Θ101584010158401015840

(1198970) sdot (119897 minus 119897

0)4

minus 1

40sinΘ(119897

0) [Θ10158401015840(1198970)]2

minus1

120cosΘ(119897

0)Θ(4)

(1198970) (119897 minus 119897

0)5

minus 1

72sinΘ(119897

0) 11987610158401015840(1198970)Θ101584010158401015840

(1198970)

minus1

720cosΘ(119897

0)Θ(5)

(1198970) (119897 minus 119897

0)6

minus 1

336cosΘ(119897

0) [Θ10158401015840(1198970)]3

+1

336sinΘ(119897

0)Θ10158401015840(1198970)Θ(4)

(1198970)

+1

504sinΘ(119897

0) [Θ101584010158401015840

(1198970)]2

minus1

5040cosΘ(119897

0)Θ(6)

(1198970) (119897 minus 119897

0)7

minus 1

384cosΘ(119897

0) [Θ10158401015840(1198970)]2

Θ101584010158401015840

(1198970)

+1

1920sinΘ(119897

0)Θ10158401015840(1198970)Θ(5)

(1198970)

+1

1152sinΘ(119897

0)Θ101584010158401015840

(1198970)Θ(4)

(1198970)

minus1

40320cosΘ(119897

0)Θ(7)

(1198970) (119897 minus 119897

0)8

+ 1

3456sinΘ(119897

0) [Θ10158401015840(1198970)]4

minus1

1296cosΘ(119897

0)Θ10158401015840(1198970) [Θ101584010158401015840

(1198970)]2

minus1

1728cosΘ(119897

0) [Θ10158401015840(1198970)]2

Θ(4)

(1198970)

minus1

12960sin119876 (119897

0) 11987610158401015840(1198970) 119876(6)

(1198970)

minus1

6480sin119876 (119897

0) 119876101584010158401015840

(1198970) 119876(5)

(1198970)

minus1

10368sinΘ(119897

0) [Θ(4)

(1198970)]2

+1

362880cosΘ(119897

0)Θ(8)

(1198970) (119897 minus 119897

0)9+ sdot sdot sdot

(45)

where

Θ(1198970) =

1

2(1198961+ 1198962) 1198970plusmn

119897119896

120587radicminus11989611198962

Θ10158401015840(1198970) = ∓

120587

119897119896

radicminus11989611198962

Θ10158401015840(1198970) = ∓

120587

119897119896

radicminus11989611198962

Θ101584010158401015840

(1198970) =

1

2

1205872

1198972

119896

(1198961+ 1198962)

Θ(4)

(1198970) = plusmn

1205873

1198973

119896

radicminus11989611198962

Θ(5)

(1198970) = minus

1

2

1205874

1198974

119896

(1198961+ 1198962)

Θ(6)

(1198970) = ∓

1205875

1198975

119896

radicminus11989611198962

Θ(7)

(1198970) =

1

2

1205876

1198976

119896

(1198961+ 1198962)

Θ(8)

(1198970) = plusmn

1205877

1198977

119896

radicminus11989611198962

(46)

In expressions with double signs plusmn and ∓ the upper onecorresponds to the case 119896

1gt 0 119896

2lt 0 while the lower one

refers to 1198961lt 0 119896

2gt 0

The value of tangent slope at the end of the transitioncurvemakes it possible to connect to it a curve of curvature 119896

2

satisfying the tangency condition of both the curves Thevalue is the same as in the previous cases It is determinedby (25)

6 Comparison of Horizontal Ordinates ofDetermined Transition Curves

61 The Shaping of Horizontal Ordinates of ComparableCurves If the analysis is carried out for railway routes thenthe necessity of meeting kinematic conditions (10) and (11)will require a diversification of the lengths of the comparedwith each other curves (using the same speed of trains)Moreover it is necessary to individually consider cases ofconnecting homogeneous curvatures (sign 119896

1= sign 119896

2) and

inverse arcs (sign 1198961

= sign 1198962)

It is assumed that at joints of consistent arcs with radii1198771= 1200m and 119877

2= 700m an attempt is made to attain a

speed V = 110 kmh Hence it is necessary to employ on arcssuperelevation of ℎ

1= 80mmand ℎ

2= 115mm respectively

The retention of the same kinematic parameters can providediversified lengths of the transition curve

(i) for a curve of linear curvature 119897119896= 40m

(ii) for a curve of polynomial curvature 119897119896= 60m

(iii) for a curve of trigonometric curvature 119897119896= 62 832m

10 Mathematical Problems in Engineering

The above length relations result directly from the equa-tions given at Sections 32 41 and 51

Figure 5 gives diagrams of horizontal ordinates for com-pared transition curves of positive and negative curvaturevalues As seen the curvature ordinates of linear curvatureevidently deviate from the ordinates of other curves In turncurves of polynomial and trigonometric curvature are inprinciple consistent A difference can only be noted in thefinal area And so the class of the function describing thecurvature plays here a significant part with respect to a curveof linear curvature it is a function of class C0 while regardingthe other two curves it is a function of class C1

At the joint of inverse arcs of radii 1198771= 1200m and 119877

2=

700m an attempt is made to reach the speed of V = 90 kmhHence it follows that it is necessary to apply superelevationalong arcs attaining ℎ

1= 20mm and ℎ

2= 45mm respec-

tively By the use the same kinematic parameters it is possibleto obtain the following lengths of the transition curve

(i) for a curve of linear curvature 119897119896= 60m

(ii) for a curve of polynomial curvature 119897119896= 90m

(iii) for a curve of trigonometric curvature 119897119896= 94 248m

Figure 6 presents diagrams of horizontal ordinates ofcompared transition curves for positive and negative valuesof curvature

As can be seen like in the case of consistent arcs the curveordinates of linear curvature deviate significantly from theordinates of the remaining curves but the shape of thesecurves also differs one from the other The class of functiondescribing the curvature is still important However in therange of the same class various geometric solutions arepossible

62 Negotiation of Transition Curves in a Geometric SystemThe transition curves do not appear alone in a geometricsystem but must be connected with some arcs in the neigh-borhood If the input arc is circular then its equation is asfollows

(i) for 1198961gt 0

119910 (119909) = 1198771minus radic1198772

1minus 1199092 (47)

(ii) for 1198961lt 0

119910 (119909) = minus (1198771minus radic1198772

1minus 1199092) (48)

where 119909 lt 0

However if the output arc is also a circular arc then it isexpressed in the following equation

(i) for 1198962gt 0

119910 (119909) = 119910119878minus radic1198772

2minus (119909119878minus 119909)2

119909119878= 119909119870

minus119904119870

radic1 + 1199042

119870

1198772 119910

119878= 119910119870

+1

radic1 + 1199042

119870

1198772

(49)

0051

152

25

0 10 20 30 40 50 60 70x (m)

y(m

)

minus05

minus1

minus15

minus2

minus25

Figure 5 Diagrams of horizontal ordinates of curves joiningconsistent arcs for positive and negative values of curvature (usingdifferent horizontal and vertical scales) curve of linear curvatureline in violet colour curve of polynomial curvature in green colourand curve of trigonometric curvature in red colour

0

05

1

15

0 20 40 60 80 100x (m)

y(m

)

minus05

minus1

minus15

Figure 6 Diagrams of horizontal ordinates of curves connectinginverse arcs for positive and negative values of curvature (usingdifferent horizontal and vertical scales) the curve of linear curvatureis in violet color the curve of polynomial curvature is in green colorand the curve of trigonometric curvature is in red color

(ii) for 1198962lt 0

119910 (119909) = 119910119878+ radic1198772

2minus (119909119878minus 119909)2

119909119878= 119909119870

+119904119870

radic1 + 1199042

119870

1198772 119910

119878= 119910119870

minus1

radic1 + 1199042

119870

1198772

(50)

where 119909 gt 119909119870

In (49) and (50) 119909119870denotes the abscissae of the transition

curve end119910119870is the ordinate of the transition curve endwhile

119904119870is the value of tangent at the end of the transition curve

Values for numerical examples under consideration are givenin Table 1

Figures 7 and 8 illustrate the geometric systems made upof two circular arcs of consistent curvature connected withvarious types of transition curves As seen the application of agiven type of transition curve leads to an adequate position ofthe output circular arcThe position of this arc by the use of alinear transition curve clearly deviates from such cases whereadvantage is taken of curves with polynomial and trigono-metrical curvatureWith regard to the latter amutual positionof the circular arc varies insignificantly

Mathematical Problems in Engineering 11

Table 1 Comparison of values 119909119870 119910119870 and 119904

119870for numerical examples analyzed

Curve 119897119896[m] Θ(119897

119896) [rad] 119904

119870119909119870[m] 119910

119870[m]

Case 1 1198961= 11200 radm and 119896

2= 1700 radm

Curve C0 40 0045238 0045269 39988 0825Curve C1 (p) 60 0067857 0067961 59963 1821Curve C1 (t) 62832 0071060 0071180 62796 1940

Case 2 1198961= 11200 radm and 119896

2= minus1700 radm

Curve C0 60 minus0017857 minus0017859 59999 0143Curve C1 (p) 90 minus0026786 minus0026792 89991 0627Curve C1 (t) 94248 minus0028050 minus0028057 94238 0714

Case 3 1198961= minus11200 radm and 119896

2= 1700 radm

Curve C0 60 001786 0017859 59999 minus0143Curve C1 (p) 90 0026786 0026792 89991 minus0627Curve C1 (t) 94248 0028050 0028057 94238 minus0714

Case 4 1198961= minus11200 radm and 119896

2= minus1700 radm

Curve C0 40 minus0045238 minus0045269 39988 minus0825Curve C1 (p) 60 minus0067857 minus0067961 59969 minus1821Curve C1 (t) 62832 minus0071060 minus0071180 62976 minus1940

0

1

2

3

4

5

6

0 20 40 60 80 100 120x (m)

y(m

)

minus40 minus20

Figure 7 Examples of geometric systems consisting of two circulararcs with consistent curvature (brown line) connected with differenttransition curves for positive values of curvature (using differenthorizontal and vertical scales) curve of linear curvature in violetline curve of polynomial curvature in green and curve of trigono-metric curvature in red

00 20 40 60 80 100 120

x (m)

y(m

)

minus40 minus20

minus1

minus2

minus3

minus4

minus5

minus6

minus7

Figure 8 Examples of geometric systems consisting of two circulararcs with consistent curvature (brown line) connected with differenttransition curves for negative values of curvature (using differenthorizontal and vertical scales) curve of linear curvature in violetline curve of polynomial curvature in green and curve of trigono-metric curvature in red

0

05

1

15

0 50 100 150x (m)

y(m

)

minus05

minus1

minus15

minus2

minus50

Figure 9 Examples of geometric systems consisting of two inversecircular arcs (brown line) connected with different types of transi-tion curves for positive values of input curvature (using differenthorizontal and vertical scales) curve of linear curvature in violetline curve of polynomial curvature in green and curve of trigono-metric curvature in red

Figures 9 and 10 illustrate geometric systems made up oftwo inverse circular arcs connected with various types oftransition curves

Here the layout of the output circular arc is much morediversifiedThe position of the arc obtained by the applicationof the transition curve of linear curvature diverts by severaldozenmeters from situationswhere curves of polynomial andtrigonometric curvature have been used The difference forcurves of polynomial and trigonometric curvature attains inthe numerical example analyzed an order of several meters

The presented examples are taken in the very nature ofthings at random They reveal wide possibilities for the pre-sented analytical method in modelling geometric systems Itsapplication makes it possible to generate diversified layout ofthe communication routes adapted to the field conditions andother circumstances Simultaneously the method ensurescorrectness of the obtained solution in view of the kinematicparameters magnitudes in force

12 Mathematical Problems in Engineering

0

05

1

15

2

0 50 100 150x (m)

y(m

)

minus05

minus1

minus15

minus50

Figure 10 Examples of geometric systems consisting of twoinverse circular arcs (brown line) connected with different typesof transition curves for negative values of input curvature (usingdifferent horizontal and vertical scales) curve of linear curvaturein violet line curve of polynomial curvature in green and curve oftrigonometric curvature in red

7 Conclusions

The presented universal method of modelling geometricalsystem of a communication route is based on the determina-tion of an adequate curvature equation by using differentialequationsThe technique enables us to design connections ofvarious geometrical elements of the route indicating diversifi-cation in its curvature It is characterized by an analytical wayof recording certain functions and moreover it is possible tospecify in advance the class of these functions

The sought-after connection is recorded in the form ofparametric equations 119909(119897) and 119910(119897) where parameter 119897 isthe position of a given point along the length of the curveThe determination of 119909(119897) and 119910(119897) needs the expansion ofintegrand functions into Maclaurin series and with regard toinverse curvatures into Taylor series

The determination of coordinates of an adequate tran-sition curve in rectangular coordinate system 119909 119910 makesit possible for an easy transfer of it to Gauss-Kruger (119883 119884)

conformal coordinates and its subsequent layout in fieldmaking use of the satellite measuring technique GNSS

The method can be applied to identify various typesof transition curves The paper gives examples of solvingthe problem involving linear change of curvature and thecurvature in polynomial and trigonometric form

The performed comparative analysis reveals a wide rangeof possibilities which are offered by the presented designmethod Its application enables us to generate diversifiedlayout of the route It simultaneously ensures a completecontrol of correctness of the obtained solutions in view ofsatisfying the geometric and kinematic conditions

The presented method of modelling curvature has auniversal character It can be used for vehicular roads andrailway lines as well In the case of railway lines therearises an additional possibility of modelling in a similar waythe superelevation ramp understood here as a determineddifference in the height of the rail courses

Conflict of Interests

The author declares that he has no conflict of interestsregarding the publication of this paper

References

[1] W Koc and C Specht ldquoApplication of the Polish active GNSSgeodetic network for surveying and design of the railroadrdquo inProceedings of the 1st International Conference on Road and RailInfrastructure (CETRA rsquo10) pp 757ndash762 May 2010

[2] P Lipar M Lakner T Maher andM Zura ldquoEstimation of roadcenterline curvature from raw GPS datardquo Baltic Journal of Roadand Bridge Engineering vol 6 no 3 pp 163ndash168 2011

[3] L Tasci and N Kuloglu ldquoInvestigation of a new transitioncurverdquoTheBaltic Journal of Road and Bridge Engineering vol 6no 1 pp 23ndash29 2011

[4] H-H Cai and G-J Wang ldquoA new method in highway routedesign joining circular arcs by a single C-Bezier curve withshape parameterrdquo Journal of Zhejiang University Science A vol10 no 4 pp 562ndash569 2009

[5] S Dimulyo Z Habib and M Sakai ldquoFair cubic transition bet-ween two circles with one circle inside or tangent to the otherrdquoNumerical Algorithms vol 51 no 4 pp 461ndash476 2009

[6] G Farin ldquoClass A Bezier curvesrdquo Computer Aided GeometricDesign vol 23 no 7 pp 573ndash581 2006

[7] Z Habib and M Sakai ldquoG2 pythagorean hodograph quintictransition between two circles with shape controlrdquo ComputerAided Geometric Design vol 24 no 5 pp 252ndash266 2007

[8] G Harary and A Tal ldquoThe natural 3D spiralrdquo Computer Graph-ics Forum vol 30 no 2 pp 237ndash246 2011

[9] K T Miura ldquoA general equation of aesthetic curves and its self-affinityrdquo Computer-Aided Design and Applications vol 3 no 1ndash4 pp 457ndash464 2006

[10] S H Yahaya M S Salleh and J M Ali ldquoSpur gear design withan S-shaped transition curve application using mathematicaand CAD toolsrdquo in Proceedings of the International Conferenceon Computer Technology and Development (ICCTD rsquo09) pp273ndash276 Dubai United Arab Emirates November 2009

[11] N Yoshida and T Saito ldquoQuasi-aesthetic curves in rationalcubic Bezier formsrdquo Computer-Aided Design and Applicationsvol 4 no 1ndash6 pp 477ndash486 2007

[12] R Ziatdinov N Yoshida and T-W Kim ldquoAnalytic parametricequations of log-aesthetic curves in terms of incomplete gammafunctionsrdquo Computer Aided Geometric Design vol 29 no 2 pp129ndash140 2012

[13] R J Grabowski ldquoSmooth curvilinear transitions along vehicu-lar roads and railway linesrdquo Science Papers of AGH Universityof Science and Technology no 82 Cracow Poland 1984(Polish)

[14] L T Klauder ldquoTrack transition curve geometry based on gegen-bauer polynomialsrdquo in Proceedings of the 6th International Con-ference on Railway Engineering May 2003

[15] W Koc Elements of Design Theory on Track Systems GdanskUniversity of Technology Gdansk Poland 2004 (Polish)

[16] E Mieloszyk and W Koc ldquoGeneral dynamic method for det-ermining transition curve equationsrdquo Rail InternationalmdashSchienen der Welt vol 10 pp 32ndash40 1991

[17] B Kufver ldquoRealigning railway in track renewalsmdashlinear versusS-shaped superelevation rampsrdquo in Proceedings of the 2ndInternational Conference on Railway Engineering May 1999

[18] W Koc and E Mieloszyk ldquoComparative analysis of selectedtransition curves by the use of dynamicmodelrdquoArchives of CivilEngineering vol 33 no 2 pp 239ndash261 1987 (Polish)

[19] S Gleason and D Gebre-Egziabher GNSS Applications andMethods Artech House Publishers Norwood NJ USA 2009

Mathematical Problems in Engineering 13

[20] B Hoffmann-Wellenhof H Lichtenegger and E Wasle GNSSGlobal Navigation Satellite SystemsmdashGPS GLONASS Galileo ampMore Springer Wien Austria 2008

[21] P Misra and P Enge Global Positioning System Signals Mea-surements and Performance Ganga-Jamuna Press LincolnMass USA 2012

[22] C Specht ldquoAccuracy and coverage of the modernized PolishMaritime differential GPS systemrdquo Advances in Space Researchvol 47 no 2 pp 221ndash228 2011

[23] W Koc and K Palikowska ldquoDynamic properties evaluation ofthe selected methods of joining route segments with differentcurvaturerdquo Technika Transportu Szynowego vol 9 pp 1785ndash1807 2012 (Polish)

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

On completing the parametric equations (20) and (23)along with (24) a correct solution of the task for the caseof diversified curvature signs is obtained (Figure 3) From apractical point of view the determination of the magnitude ofthe tangent slope angle at the end of the transition curve isvery important It amounts to

Θ(119897119896) =

1

2(1198961+ 1198962) 119897119896 (25)

A knowledge of Θ(119897119896) makes it possible to add the tran-

sition curve to the other one having curvature 1198962at its initial

point (satisfying the tangency condition of both the curves)

4 Transition Curve of Polynomial Curvature

From the point of view of vehiclesrsquo dynamics the transitioncurve of linear change of curvature is not the most advan-tageous solution [23] A definitely better one is nonlinearsolution whose characteristics are interesting enough to becomparedwith the Bezier curves preferable lately in literature[4 7]

The differential equation (2) allows to appoint an unlim-ited number of curves with nonlinear curvature change Inthe case of geometric layout of communication routes it willbe appropriate to consider the transition curves of poly-nomial and trigonometric curvature In fact such forms ofcurves are used in the standard ways to connect a straight linewith a circular arc

41 Formulation of the Curvature Equation To formulate thecurvature equation in the polynomial form let us assume thefollowing boundary conditions

119896 (0+) = 1198961 119896 (119897

minus

119896) = 1198962

1198961015840(0+) = 0 119896

1015840(119897minus

119896) = 0

(26)

and the differential equation

119896(4)

(119897) = 0 (27)

After solving the differential problem (26) (27) thefollowing formula for curvature 119896(119897) is obtained

119896 (119897) = 1198961+

3

1198972

119896

(1198962minus 1198961) 1198972minus

2

1198973

119896

(1198962minus 1198961) 1198973 (28)

The form of function Θ(119897) determined on the basis of (5)and (28) is as follows

Θ (119897) = 1198961119897 +

1

1198972

119896

(1198962minus 1198961) 1198973minus

1

21198973

119896

(1198962minus 1198961) 1198974 (29)

On rail routes the length of the transition curve isdetermined like at Section 32 for solution of linear curvatureThe following conditions should be fulfilled here

119897120595

119896ge

3

2

V120595dop

10038161003816100381610038161198862 minus 1198861

1003816100381610038161003816

119897119891

119896ge

3

2

V119891dop

1003816100381610038161003816ℎ2 minus ℎ1

1003816100381610038161003816

(30)

0

01

02

03

04

0 20 40 60 80x (m)

y(m

)

Figure 3 Diagram of horizontal ordinates 119910(119909) defined by (20)(23) and (24) for the joint of circular arcs of curvatures 119896

1=

11200 radm and 1198962

= minus1700 radm with transition curve oflength 119897

119896= 60m (using different horizontal and vertical scales)

The adopted length of the transition curve must satisfycondition (19)

42 Coordinates of Transition Curve Connecting Uniform Cur-vatures On expanding functions cosΘ(119897) and sinΘ(119897) intoMaclaurin series and after integration the following paramet-ric equations are acquired

119909 (119897) = int cosΘ (119897) 119889119897

= 119897 minus1198962

1

61198973+ [

1198964

1

120minus

1198961

51198972

119896

(1198962minus 1198961)] 1198975

+ [1198961

121198973

119896

(1198962minus 1198961)] 1198976

minus [1198966

1

5040minus

1198963

1

421198972

119896

(1198962minus 1198961) +

1

141198974

119896

(1198962minus 1198961)2] 1198977

minus [1198963

1

961198973

119896

(1198962minus 1198961) minus

1

161198975

119896

(1198962minus 1198961)2] 1198978

+ [1198968

1

362880minus

1198965

1

10801198972

119896

(1198962minus 1198961) +

1198962

1

361198974

119896

(1198962minus 1198961)2

+1

721198976

119896

(1198962minus 1198961)2] 1198979+ sdot sdot sdot

119910 (119897) = int sinΘ (119897) 119889119897

=1198961

21198972minus [

1198963

1

24minus

1

41198972

119896

(1198962minus 1198961)] 1198974

minus1

101198973

119896

(1198962minus 1198961) 1198975+ [

1198965

1

720minus

1198962

1

121198972

119896

(1198962minus 1198961)] 1198976

+1198962

1

281198973

119896

(1198962minus 1198961) 1198977

6 Mathematical Problems in Engineering

0

00005

0001

0 20 40 60 80 100l (m)minus00005

minus0001

minus00015

minus0002

k(radm

)

Figure 4 Diagram of curvature 119896(119897) described by (28) for theconnection of circular arcs of curvatures 119896

1= 11200 radm and

1198962

= minus1700 radm with the use of transition curve of length 119897119896

=

90m

minus [1198967

1

40320minus

1198964

1

1921198972

119896

(1198962minus 1198961) +

1198961

161198974

119896

(1198962minus 1198961)2] 1198978

+ [1198961

181198975

119896

(1198962minus 1198961)2minus

1198964

1

4321198973

119896

(1198962minus 1198961)] 1198979+ sdot sdot sdot

(31)

Equations (31) for 1198961

= 0 describe the so-called Blosscurve which can be used to join a straight to a circular arc

43 Coordinates of Transition Curve Connecting Inverse Cur-vatures When the arcs are inverse (ie the curvature signsare different Figure 4) there occurs a problem relating to thedetermined coordinates 119909(119897) and 119910(119897) which results fromfunction Θ(119897) For sign 119896

1= sign 119896

2function Θ(119897) is a mono-

tonic function while for sign 1198961

= sign 1198962in the diagram of

function Θ(119897) there appears extremum at point 1198970 where

Θ1015840(1198970) = 119896 (119897

0) = 1198961+

3

1198972

119896

(1198962minus 1198961) 1198972

0minus

2

1198973

119896

(1198962minus 1198961) 1198973

0= 0

(32)

Parametric equations of transition curve (31) are valid for119897 isin ⟨0 119897

0⟩The determination of value 119897

0makes it necessary to

solve (32) which is cubic The sought-after value 1198970

isin ⟨0 119897119896⟩

satisfying conditions of the problem is defined by the formula

1198970= [

1

2minus cos(

120593

3+

120587

3)] 119897119896 (33)

where angle 120593 follows from the relation

cos120593 =1198961+ 1198962

1198962minus 1198961

(34)

Under such circumstances in order to determine para-metric equations of the transition curve for 119897 isin (119897

0 119897119896⟩ func-

tions cosΘ(119897) and sinΘ(119897) should be expanded into Taylor

series after integration the following parametric equations ofthe transition curve are obtained

119909 (119897) = int cosΘ (119897) 119889119897

= 119909 (1198970) + cosΘ(119897

0) (119897 minus 119897

0)

minus1

6sinΘ(119897

0)Θ10158401015840(1198970) (119897 minus 119897

0)3

minus1

24sinΘ(119897

0)Θ101584010158401015840

(1198970) (119897 minus 119897

0)4

minus 1

40cosΘ(119897

0) [Θ10158401015840(1198970)]2

+1

120sinΘ(119897

0)Θ(4)

(1198970) (119897 minus 119897

0)5

minus1

72cosΘ(119897

0)Θ10158401015840(1198970)Θ101584010158401015840

(1198970) (119897 minus 119897

0)6

+ 1

336sinΘ(119897

0) [Θ10158401015840(1198970)]3

minus1

336cosΘ(119897

0)Θ10158401015840(1198970)Θ(4)

(1198970)

minus1

504cosΘ(119897

0) [Θ101584010158401015840

(1198970)]2

(119897 minus 1198970)7

+ 1

384sinΘ(119897

0) [Θ10158401015840(1198970)]2

Θ101584010158401015840

(1198970)

minus1

1152cosΘ(119897

0)Θ101584010158401015840

(1198970)Θ(4)

(1198970) (119897 minus 119897

0)8

+ 1

3456cosΘ(119897

0) [Θ10158401015840(1198970)]4

+1

1296sinΘ(119897

0)Θ10158401015840(1198970) [Θ101584010158401015840

(1198970)]2

+1

1728sinΘ(119897

0) [Θ10158401015840(1198970)]2

Θ(4)

(1198970)

minus1

10368cosΘ(119897

0) [Θ(4)

(1198970)]2

(119897 minus 1198970)9+ sdot sdot sdot

119910 (119897) = int sinΘ (119897) 119889119897

= 119910 (1198970) + sinΘ(119897

0) (119897 minus 119897

0)

+1

6[cosΘ(119897

0)Θ10158401015840(1198970)] (119897 minus 119897

0)3

+1

24[cosΘ(119897

0)Θ101584010158401015840

(1198970)] (119897 minus 119897

0)4

minus 1

40sinΘ(119897

0) [Θ10158401015840(1198970)]2

minus1

120cosΘ(119897

0)Θ(4)

(1198970) (119897 minus 119897

0)5

minus1

72[sinΘ(119897

0)Θ10158401015840(1198970)Θ101584010158401015840

(1198970)] (119897 minus 119897

0)6

Mathematical Problems in Engineering 7

minus 1

336cosΘ(119897

0) [Θ10158401015840(1198970)]3

+1

336sinΘ(119897

0)Θ10158401015840(1198970)Θ(4)

(1198970)

+1

504sinΘ(119897

0) [Θ101584010158401015840

(1198970)]2

(119897 minus 1198970)7

minus 1

384cosΘ(119897

0) [Θ10158401015840(1198970)]2

Θ101584010158401015840

(1198970)

+1

1152sinΘ(119897

0)Θ101584010158401015840

(1198970)Θ(4)

(1198970) (119897 minus 119897

0)8

+ 1

3456sinΘ(119897

0) [Θ10158401015840(1198970)]4

minus1

1296cosΘ(119897

0)Θ10158401015840(1198970) [Θ101584010158401015840

(1198970)]2

minus1

1728cosΘ(119897

0) [Θ10158401015840(1198970)]2

Θ(4)

(1198970)

minus1

10368sinΘ(119897

0) [Θ(4)

(1198970)]2

(119897 minus 1198970)9+ sdot sdot sdot

(35)

where

Θ(1198970) = 11989611198970+

1

1198972

119896

(1198962minus 1198961) 1198973

0minus

1

21198973

119896

(1198962minus 1198961) 1198974

0

Θ10158401015840(1198970) =

6

1198972

119896

(1198962minus 1198961) 1198970minus

6

1198973

119896

(1198962minus 1198961) 1198972

0

Θ101584010158401015840

(1198970) =

6

1198972

119896

(1198962minus 1198961) minus

12

1198973

119896

(1198962minus 1198961) 1198970

Θ(4)

(1198970) = minus

12

1198973

119896

(1198962minus 1198961)

(36)

Making use of parametric equations (31) and also (35) onecan obtain a correct solution of the problem with diversifiedsigns of curvature

Value of the tangent slope at the end of the transitioncurvemakes it possible to connect to it a curve of curvature 119896

2

satisfying simultaneously the tangency condition of both thecurves The above value is the same as in the case of linearcurvature It is defined by (25)

5 Transition Curve of Curvature inTrigonometric Form

51 Formulation of Curvature Equation As before boundaryconditions (26) and another differential equation areassumed

119896(4)

(119897) +1205872

1198972

119896

11989610158401015840(119897) = 0 (37)

As soon as the differential problem (26) (37) has beensolved a formula for curvature is obtained

119896 (119897) =1

2(1198961+ 1198962) minus

1

2(1198962minus 1198961) cos 120587

119897119896

119897 (38)

The form of function Θ(119897) determined by the use of (5)and (38) is as follows

Θ (119897) =1

2(1198961+ 1198962) 119897 minus

1

2(1198962minus 1198961)119897119896

120587sin 120587

119897119896

119897 (39)

The length of the transition curve on railway routesfollows from conditions

119897120595

119896ge

120587

2

V120595dop

10038161003816100381610038161198862 minus 1198861

1003816100381610038161003816

119897119891

119896ge

120587

2

V119891dop

1003816100381610038161003816ℎ2 minus ℎ1

1003816100381610038161003816

(40)

The assumed length of the curve must satisfy the condi-tion (19)

52 Ordinates of Transition Curve Connecting UniformCurva-tures After the expansion of functions cosΘ(119897) and sinΘ(119897)

intoMaclaurin series to be next integrated the following para-metric equations are obtained

119909 (119897) = int cosΘ (119897) 119889119897

= 119897 minus1198962

1

61198973+ [

1198964

1

120minus

1

60

1205872

1198972

119896

1198961(1198962minus 1198961)] 1198975

minus [1198966

1

5040minus

1

504

1205872

1198972

119896

1198963

1(1198962minus 1198961)

minus1

1680

1205874

1198974

119896

1198961(1198962minus 1198961) minus

1

2016

1205874

1198974

119896

(1198962minus 1198961)2] 1198977

+ [1198968

1

362880minus

1

12960

1205872

1198972

119896

1198965

1(1198962minus 1198961)

minus1

5184

1205874

1198974

119896

1198962

1(1198962minus 1198961)2

minus1

12960

1205874

1198974

119896

1198963

1(1198962minus 1198961) minus

1

90720

1205876

1198976

119896

1198961(1198962minus 1198961)

+1

25920

1205876

1198976

119896

(1198962minus 1198961)2] 1198979+ sdot sdot sdot

8 Mathematical Problems in Engineering

119910 (119897) = int sinΘ (119897) 119889119897

=1

211989611198972minus [

1

241198963

1minus

1

48

1205872

1198972

119896

(1198962minus 1198961)] 1198974

+ [1

7201198965

1minus

1

1441198962

1

1205872

1198972

119896

(1198962minus 1198961)

minus1

1440

1205874

1198974

119896

(1198962minus 1198961)] 1198976

minus [1

403201198967

1minus

1

23041198964

1

1205872

1198972

119896

(1198962minus 1198961)

+1

2304

1205874

1198974

119896

1198961(1198962minus 1198961)2minus

1

3840

1205874

1198974

119896

1198962

1(1198962minus 1198961)

+1

80640

1205876

1198976

119896

(1198962minus 1198961)] 1198978+ sdot sdot sdot

(41)

Equations (41) for 1198961

= 0 describe the curve known ascosinusoid which can be used to connect a straight with acircular arc

53 Transition Curve Coordinates Connecting Inverse Curva-tures Near inverse arcs in the diagram of functionΘ(119897) thereappears the extremum at point 119897

0 where

Θ1015840(1198970) = 119896 (119897

0) =

1

2(1198961+ 1198962) minus

1

2(1198962minus 1198961) cos 120587

119897119896

1198970= 0

(42)

The transition curve parametric equations (41) are effec-tive for 119897 isin ⟨0 119897

0⟩ To determine value 119897

0one must solve (42)

The equation can be written in the form

cos 120587

119897119896

1198970=

1198961+ 1198962

1198962minus 1198961

(43)

It is easy to prove that for sign 1198961

= sign 1198962it is necessary

to satisfy the condition |1198961+1198962| lt |1198962minus1198961| thus |(119896

1+1198962)(1198962minus

1198961)| lt 1 The formula for length 119897

0is as follows

1198970=

1

120587(arccos1198961 + 119896

2

1198962minus 1198961

) 119897119896 (44)

For 119897 isin (1198970 119897119896⟩ functions cosΘ(119897) and sinΘ(119897) should

be expanded into Taylor series The integration provides thefollowing equations

119909 (119897) = int cosΘ (119897) 119889119897

= 119909 (1198970) + cosΘ(119897

0) (119897 minus 119897

0)

minus1

6sinΘ(119897

0)Θ10158401015840(1198970) (119897 minus 119897

0)3

minus1

24sinΘ(119897

0)Θ101584010158401015840

(1198970) sdot (119897 minus 119897

0)4

minus 1

40cosΘ(119897

0) [Θ10158401015840(1198970)]2

+1

120sinΘ(119897

0)Θ(4)

(1198970) (119897 minus 119897

0)5

minus 1

72cosΘ(119897

0)Θ10158401015840(1198970)Θ101584010158401015840

(1198970)

+1

720sinΘ(119897

0)Θ(5)

(1198970) (119897 minus 119897

0)6

+ 1

336sinΘ(119897

0) [Θ10158401015840(1198970)]3

minus1

336cosΘ(119897

0)Θ10158401015840(1198970)Θ(4)

(1198970)

minus1

504cosΘ(119897

0) [Θ101584010158401015840

(1198970)]2

minus1

5040sinΘ(119897

0)Θ(6)

(1198970) (119897 minus 119897

0)7

+ 1

384sinΘ(119897

0) [Θ10158401015840(1198970)]2

Θ101584010158401015840

(1198970)

minus1

1920cosΘ(119897

0)Θ10158401015840(119897) Θ(5)

(119897)

minus1

1152cosΘ(119897

0)Θ101584010158401015840

(1198970)Θ(4)

(1198970)

minus1

40320sinΘ(119897

0)Θ(7)

(1198970) (119897 minus 119897

0)8

+ 1

3456cosΘ(119897

0) [Θ10158401015840(1198970)]4

+1

1296sinΘ(119897

0)Θ10158401015840(1198970) [Θ101584010158401015840

(1198970)]2

+1

1728sinΘ(119897

0) [Θ10158401015840(1198970)]2

Θ(4)

(1198970)

minus1

12960cosΘ(119897

0)Θ10158401015840(1198970)Θ(6)

(1198970)

minus1

6480cosΘ(119897

0)Θ101584010158401015840

(1198970)Θ(5)

(1198970)

minus1

10368cosΘ(119897

0) [Θ(4)

(1198970)]2

minus1

362880sinΘ(119897

0)Θ(8)

(1198970) (119897 minus 119897

0)9+ sdot sdot sdot

Mathematical Problems in Engineering 9

119910 (119897) = int sinΘ (119897) 119889119897

= 119910 (1198970) + sinΘ(119897

0) (119897 minus 119897

0)

+1

6cosΘ(119897

0)Θ10158401015840(1198970) (119897 minus 119897

0)3

+1

24cosΘ(119897

0)Θ101584010158401015840

(1198970) sdot (119897 minus 119897

0)4

minus 1

40sinΘ(119897

0) [Θ10158401015840(1198970)]2

minus1

120cosΘ(119897

0)Θ(4)

(1198970) (119897 minus 119897

0)5

minus 1

72sinΘ(119897

0) 11987610158401015840(1198970)Θ101584010158401015840

(1198970)

minus1

720cosΘ(119897

0)Θ(5)

(1198970) (119897 minus 119897

0)6

minus 1

336cosΘ(119897

0) [Θ10158401015840(1198970)]3

+1

336sinΘ(119897

0)Θ10158401015840(1198970)Θ(4)

(1198970)

+1

504sinΘ(119897

0) [Θ101584010158401015840

(1198970)]2

minus1

5040cosΘ(119897

0)Θ(6)

(1198970) (119897 minus 119897

0)7

minus 1

384cosΘ(119897

0) [Θ10158401015840(1198970)]2

Θ101584010158401015840

(1198970)

+1

1920sinΘ(119897

0)Θ10158401015840(1198970)Θ(5)

(1198970)

+1

1152sinΘ(119897

0)Θ101584010158401015840

(1198970)Θ(4)

(1198970)

minus1

40320cosΘ(119897

0)Θ(7)

(1198970) (119897 minus 119897

0)8

+ 1

3456sinΘ(119897

0) [Θ10158401015840(1198970)]4

minus1

1296cosΘ(119897

0)Θ10158401015840(1198970) [Θ101584010158401015840

(1198970)]2

minus1

1728cosΘ(119897

0) [Θ10158401015840(1198970)]2

Θ(4)

(1198970)

minus1

12960sin119876 (119897

0) 11987610158401015840(1198970) 119876(6)

(1198970)

minus1

6480sin119876 (119897

0) 119876101584010158401015840

(1198970) 119876(5)

(1198970)

minus1

10368sinΘ(119897

0) [Θ(4)

(1198970)]2

+1

362880cosΘ(119897

0)Θ(8)

(1198970) (119897 minus 119897

0)9+ sdot sdot sdot

(45)

where

Θ(1198970) =

1

2(1198961+ 1198962) 1198970plusmn

119897119896

120587radicminus11989611198962

Θ10158401015840(1198970) = ∓

120587

119897119896

radicminus11989611198962

Θ10158401015840(1198970) = ∓

120587

119897119896

radicminus11989611198962

Θ101584010158401015840

(1198970) =

1

2

1205872

1198972

119896

(1198961+ 1198962)

Θ(4)

(1198970) = plusmn

1205873

1198973

119896

radicminus11989611198962

Θ(5)

(1198970) = minus

1

2

1205874

1198974

119896

(1198961+ 1198962)

Θ(6)

(1198970) = ∓

1205875

1198975

119896

radicminus11989611198962

Θ(7)

(1198970) =

1

2

1205876

1198976

119896

(1198961+ 1198962)

Θ(8)

(1198970) = plusmn

1205877

1198977

119896

radicminus11989611198962

(46)

In expressions with double signs plusmn and ∓ the upper onecorresponds to the case 119896

1gt 0 119896

2lt 0 while the lower one

refers to 1198961lt 0 119896

2gt 0

The value of tangent slope at the end of the transitioncurvemakes it possible to connect to it a curve of curvature 119896

2

satisfying the tangency condition of both the curves Thevalue is the same as in the previous cases It is determinedby (25)

6 Comparison of Horizontal Ordinates ofDetermined Transition Curves

61 The Shaping of Horizontal Ordinates of ComparableCurves If the analysis is carried out for railway routes thenthe necessity of meeting kinematic conditions (10) and (11)will require a diversification of the lengths of the comparedwith each other curves (using the same speed of trains)Moreover it is necessary to individually consider cases ofconnecting homogeneous curvatures (sign 119896

1= sign 119896

2) and

inverse arcs (sign 1198961

= sign 1198962)

It is assumed that at joints of consistent arcs with radii1198771= 1200m and 119877

2= 700m an attempt is made to attain a

speed V = 110 kmh Hence it is necessary to employ on arcssuperelevation of ℎ

1= 80mmand ℎ

2= 115mm respectively

The retention of the same kinematic parameters can providediversified lengths of the transition curve

(i) for a curve of linear curvature 119897119896= 40m

(ii) for a curve of polynomial curvature 119897119896= 60m

(iii) for a curve of trigonometric curvature 119897119896= 62 832m

10 Mathematical Problems in Engineering

The above length relations result directly from the equa-tions given at Sections 32 41 and 51

Figure 5 gives diagrams of horizontal ordinates for com-pared transition curves of positive and negative curvaturevalues As seen the curvature ordinates of linear curvatureevidently deviate from the ordinates of other curves In turncurves of polynomial and trigonometric curvature are inprinciple consistent A difference can only be noted in thefinal area And so the class of the function describing thecurvature plays here a significant part with respect to a curveof linear curvature it is a function of class C0 while regardingthe other two curves it is a function of class C1

At the joint of inverse arcs of radii 1198771= 1200m and 119877

2=

700m an attempt is made to reach the speed of V = 90 kmhHence it follows that it is necessary to apply superelevationalong arcs attaining ℎ

1= 20mm and ℎ

2= 45mm respec-

tively By the use the same kinematic parameters it is possibleto obtain the following lengths of the transition curve

(i) for a curve of linear curvature 119897119896= 60m

(ii) for a curve of polynomial curvature 119897119896= 90m

(iii) for a curve of trigonometric curvature 119897119896= 94 248m

Figure 6 presents diagrams of horizontal ordinates ofcompared transition curves for positive and negative valuesof curvature

As can be seen like in the case of consistent arcs the curveordinates of linear curvature deviate significantly from theordinates of the remaining curves but the shape of thesecurves also differs one from the other The class of functiondescribing the curvature is still important However in therange of the same class various geometric solutions arepossible

62 Negotiation of Transition Curves in a Geometric SystemThe transition curves do not appear alone in a geometricsystem but must be connected with some arcs in the neigh-borhood If the input arc is circular then its equation is asfollows

(i) for 1198961gt 0

119910 (119909) = 1198771minus radic1198772

1minus 1199092 (47)

(ii) for 1198961lt 0

119910 (119909) = minus (1198771minus radic1198772

1minus 1199092) (48)

where 119909 lt 0

However if the output arc is also a circular arc then it isexpressed in the following equation

(i) for 1198962gt 0

119910 (119909) = 119910119878minus radic1198772

2minus (119909119878minus 119909)2

119909119878= 119909119870

minus119904119870

radic1 + 1199042

119870

1198772 119910

119878= 119910119870

+1

radic1 + 1199042

119870

1198772

(49)

0051

152

25

0 10 20 30 40 50 60 70x (m)

y(m

)

minus05

minus1

minus15

minus2

minus25

Figure 5 Diagrams of horizontal ordinates of curves joiningconsistent arcs for positive and negative values of curvature (usingdifferent horizontal and vertical scales) curve of linear curvatureline in violet colour curve of polynomial curvature in green colourand curve of trigonometric curvature in red colour

0

05

1

15

0 20 40 60 80 100x (m)

y(m

)

minus05

minus1

minus15

Figure 6 Diagrams of horizontal ordinates of curves connectinginverse arcs for positive and negative values of curvature (usingdifferent horizontal and vertical scales) the curve of linear curvatureis in violet color the curve of polynomial curvature is in green colorand the curve of trigonometric curvature is in red color

(ii) for 1198962lt 0

119910 (119909) = 119910119878+ radic1198772

2minus (119909119878minus 119909)2

119909119878= 119909119870

+119904119870

radic1 + 1199042

119870

1198772 119910

119878= 119910119870

minus1

radic1 + 1199042

119870

1198772

(50)

where 119909 gt 119909119870

In (49) and (50) 119909119870denotes the abscissae of the transition

curve end119910119870is the ordinate of the transition curve endwhile

119904119870is the value of tangent at the end of the transition curve

Values for numerical examples under consideration are givenin Table 1

Figures 7 and 8 illustrate the geometric systems made upof two circular arcs of consistent curvature connected withvarious types of transition curves As seen the application of agiven type of transition curve leads to an adequate position ofthe output circular arcThe position of this arc by the use of alinear transition curve clearly deviates from such cases whereadvantage is taken of curves with polynomial and trigono-metrical curvatureWith regard to the latter amutual positionof the circular arc varies insignificantly

Mathematical Problems in Engineering 11

Table 1 Comparison of values 119909119870 119910119870 and 119904

119870for numerical examples analyzed

Curve 119897119896[m] Θ(119897

119896) [rad] 119904

119870119909119870[m] 119910

119870[m]

Case 1 1198961= 11200 radm and 119896

2= 1700 radm

Curve C0 40 0045238 0045269 39988 0825Curve C1 (p) 60 0067857 0067961 59963 1821Curve C1 (t) 62832 0071060 0071180 62796 1940

Case 2 1198961= 11200 radm and 119896

2= minus1700 radm

Curve C0 60 minus0017857 minus0017859 59999 0143Curve C1 (p) 90 minus0026786 minus0026792 89991 0627Curve C1 (t) 94248 minus0028050 minus0028057 94238 0714

Case 3 1198961= minus11200 radm and 119896

2= 1700 radm

Curve C0 60 001786 0017859 59999 minus0143Curve C1 (p) 90 0026786 0026792 89991 minus0627Curve C1 (t) 94248 0028050 0028057 94238 minus0714

Case 4 1198961= minus11200 radm and 119896

2= minus1700 radm

Curve C0 40 minus0045238 minus0045269 39988 minus0825Curve C1 (p) 60 minus0067857 minus0067961 59969 minus1821Curve C1 (t) 62832 minus0071060 minus0071180 62976 minus1940

0

1

2

3

4

5

6

0 20 40 60 80 100 120x (m)

y(m

)

minus40 minus20

Figure 7 Examples of geometric systems consisting of two circulararcs with consistent curvature (brown line) connected with differenttransition curves for positive values of curvature (using differenthorizontal and vertical scales) curve of linear curvature in violetline curve of polynomial curvature in green and curve of trigono-metric curvature in red

00 20 40 60 80 100 120

x (m)

y(m

)

minus40 minus20

minus1

minus2

minus3

minus4

minus5

minus6

minus7

Figure 8 Examples of geometric systems consisting of two circulararcs with consistent curvature (brown line) connected with differenttransition curves for negative values of curvature (using differenthorizontal and vertical scales) curve of linear curvature in violetline curve of polynomial curvature in green and curve of trigono-metric curvature in red

0

05

1

15

0 50 100 150x (m)

y(m

)

minus05

minus1

minus15

minus2

minus50

Figure 9 Examples of geometric systems consisting of two inversecircular arcs (brown line) connected with different types of transi-tion curves for positive values of input curvature (using differenthorizontal and vertical scales) curve of linear curvature in violetline curve of polynomial curvature in green and curve of trigono-metric curvature in red

Figures 9 and 10 illustrate geometric systems made up oftwo inverse circular arcs connected with various types oftransition curves

Here the layout of the output circular arc is much morediversifiedThe position of the arc obtained by the applicationof the transition curve of linear curvature diverts by severaldozenmeters from situationswhere curves of polynomial andtrigonometric curvature have been used The difference forcurves of polynomial and trigonometric curvature attains inthe numerical example analyzed an order of several meters

The presented examples are taken in the very nature ofthings at random They reveal wide possibilities for the pre-sented analytical method in modelling geometric systems Itsapplication makes it possible to generate diversified layout ofthe communication routes adapted to the field conditions andother circumstances Simultaneously the method ensurescorrectness of the obtained solution in view of the kinematicparameters magnitudes in force

12 Mathematical Problems in Engineering

0

05

1

15

2

0 50 100 150x (m)

y(m

)

minus05

minus1

minus15

minus50

Figure 10 Examples of geometric systems consisting of twoinverse circular arcs (brown line) connected with different typesof transition curves for negative values of input curvature (usingdifferent horizontal and vertical scales) curve of linear curvaturein violet line curve of polynomial curvature in green and curve oftrigonometric curvature in red

7 Conclusions

The presented universal method of modelling geometricalsystem of a communication route is based on the determina-tion of an adequate curvature equation by using differentialequationsThe technique enables us to design connections ofvarious geometrical elements of the route indicating diversifi-cation in its curvature It is characterized by an analytical wayof recording certain functions and moreover it is possible tospecify in advance the class of these functions

The sought-after connection is recorded in the form ofparametric equations 119909(119897) and 119910(119897) where parameter 119897 isthe position of a given point along the length of the curveThe determination of 119909(119897) and 119910(119897) needs the expansion ofintegrand functions into Maclaurin series and with regard toinverse curvatures into Taylor series

The determination of coordinates of an adequate tran-sition curve in rectangular coordinate system 119909 119910 makesit possible for an easy transfer of it to Gauss-Kruger (119883 119884)

conformal coordinates and its subsequent layout in fieldmaking use of the satellite measuring technique GNSS

The method can be applied to identify various typesof transition curves The paper gives examples of solvingthe problem involving linear change of curvature and thecurvature in polynomial and trigonometric form

The performed comparative analysis reveals a wide rangeof possibilities which are offered by the presented designmethod Its application enables us to generate diversifiedlayout of the route It simultaneously ensures a completecontrol of correctness of the obtained solutions in view ofsatisfying the geometric and kinematic conditions

The presented method of modelling curvature has auniversal character It can be used for vehicular roads andrailway lines as well In the case of railway lines therearises an additional possibility of modelling in a similar waythe superelevation ramp understood here as a determineddifference in the height of the rail courses

Conflict of Interests

The author declares that he has no conflict of interestsregarding the publication of this paper

References

[1] W Koc and C Specht ldquoApplication of the Polish active GNSSgeodetic network for surveying and design of the railroadrdquo inProceedings of the 1st International Conference on Road and RailInfrastructure (CETRA rsquo10) pp 757ndash762 May 2010

[2] P Lipar M Lakner T Maher andM Zura ldquoEstimation of roadcenterline curvature from raw GPS datardquo Baltic Journal of Roadand Bridge Engineering vol 6 no 3 pp 163ndash168 2011

[3] L Tasci and N Kuloglu ldquoInvestigation of a new transitioncurverdquoTheBaltic Journal of Road and Bridge Engineering vol 6no 1 pp 23ndash29 2011

[4] H-H Cai and G-J Wang ldquoA new method in highway routedesign joining circular arcs by a single C-Bezier curve withshape parameterrdquo Journal of Zhejiang University Science A vol10 no 4 pp 562ndash569 2009

[5] S Dimulyo Z Habib and M Sakai ldquoFair cubic transition bet-ween two circles with one circle inside or tangent to the otherrdquoNumerical Algorithms vol 51 no 4 pp 461ndash476 2009

[6] G Farin ldquoClass A Bezier curvesrdquo Computer Aided GeometricDesign vol 23 no 7 pp 573ndash581 2006

[7] Z Habib and M Sakai ldquoG2 pythagorean hodograph quintictransition between two circles with shape controlrdquo ComputerAided Geometric Design vol 24 no 5 pp 252ndash266 2007

[8] G Harary and A Tal ldquoThe natural 3D spiralrdquo Computer Graph-ics Forum vol 30 no 2 pp 237ndash246 2011

[9] K T Miura ldquoA general equation of aesthetic curves and its self-affinityrdquo Computer-Aided Design and Applications vol 3 no 1ndash4 pp 457ndash464 2006

[10] S H Yahaya M S Salleh and J M Ali ldquoSpur gear design withan S-shaped transition curve application using mathematicaand CAD toolsrdquo in Proceedings of the International Conferenceon Computer Technology and Development (ICCTD rsquo09) pp273ndash276 Dubai United Arab Emirates November 2009

[11] N Yoshida and T Saito ldquoQuasi-aesthetic curves in rationalcubic Bezier formsrdquo Computer-Aided Design and Applicationsvol 4 no 1ndash6 pp 477ndash486 2007

[12] R Ziatdinov N Yoshida and T-W Kim ldquoAnalytic parametricequations of log-aesthetic curves in terms of incomplete gammafunctionsrdquo Computer Aided Geometric Design vol 29 no 2 pp129ndash140 2012

[13] R J Grabowski ldquoSmooth curvilinear transitions along vehicu-lar roads and railway linesrdquo Science Papers of AGH Universityof Science and Technology no 82 Cracow Poland 1984(Polish)

[14] L T Klauder ldquoTrack transition curve geometry based on gegen-bauer polynomialsrdquo in Proceedings of the 6th International Con-ference on Railway Engineering May 2003

[15] W Koc Elements of Design Theory on Track Systems GdanskUniversity of Technology Gdansk Poland 2004 (Polish)

[16] E Mieloszyk and W Koc ldquoGeneral dynamic method for det-ermining transition curve equationsrdquo Rail InternationalmdashSchienen der Welt vol 10 pp 32ndash40 1991

[17] B Kufver ldquoRealigning railway in track renewalsmdashlinear versusS-shaped superelevation rampsrdquo in Proceedings of the 2ndInternational Conference on Railway Engineering May 1999

[18] W Koc and E Mieloszyk ldquoComparative analysis of selectedtransition curves by the use of dynamicmodelrdquoArchives of CivilEngineering vol 33 no 2 pp 239ndash261 1987 (Polish)

[19] S Gleason and D Gebre-Egziabher GNSS Applications andMethods Artech House Publishers Norwood NJ USA 2009

Mathematical Problems in Engineering 13

[20] B Hoffmann-Wellenhof H Lichtenegger and E Wasle GNSSGlobal Navigation Satellite SystemsmdashGPS GLONASS Galileo ampMore Springer Wien Austria 2008

[21] P Misra and P Enge Global Positioning System Signals Mea-surements and Performance Ganga-Jamuna Press LincolnMass USA 2012

[22] C Specht ldquoAccuracy and coverage of the modernized PolishMaritime differential GPS systemrdquo Advances in Space Researchvol 47 no 2 pp 221ndash228 2011

[23] W Koc and K Palikowska ldquoDynamic properties evaluation ofthe selected methods of joining route segments with differentcurvaturerdquo Technika Transportu Szynowego vol 9 pp 1785ndash1807 2012 (Polish)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

0

00005

0001

0 20 40 60 80 100l (m)minus00005

minus0001

minus00015

minus0002

k(radm

)

Figure 4 Diagram of curvature 119896(119897) described by (28) for theconnection of circular arcs of curvatures 119896

1= 11200 radm and

1198962

= minus1700 radm with the use of transition curve of length 119897119896

=

90m

minus [1198967

1

40320minus

1198964

1

1921198972

119896

(1198962minus 1198961) +

1198961

161198974

119896

(1198962minus 1198961)2] 1198978

+ [1198961

181198975

119896

(1198962minus 1198961)2minus

1198964

1

4321198973

119896

(1198962minus 1198961)] 1198979+ sdot sdot sdot

(31)

Equations (31) for 1198961

= 0 describe the so-called Blosscurve which can be used to join a straight to a circular arc

43 Coordinates of Transition Curve Connecting Inverse Cur-vatures When the arcs are inverse (ie the curvature signsare different Figure 4) there occurs a problem relating to thedetermined coordinates 119909(119897) and 119910(119897) which results fromfunction Θ(119897) For sign 119896

1= sign 119896

2function Θ(119897) is a mono-

tonic function while for sign 1198961

= sign 1198962in the diagram of

function Θ(119897) there appears extremum at point 1198970 where

Θ1015840(1198970) = 119896 (119897

0) = 1198961+

3

1198972

119896

(1198962minus 1198961) 1198972

0minus

2

1198973

119896

(1198962minus 1198961) 1198973

0= 0

(32)

Parametric equations of transition curve (31) are valid for119897 isin ⟨0 119897

0⟩The determination of value 119897

0makes it necessary to

solve (32) which is cubic The sought-after value 1198970

isin ⟨0 119897119896⟩

satisfying conditions of the problem is defined by the formula

1198970= [

1

2minus cos(

120593

3+

120587

3)] 119897119896 (33)

where angle 120593 follows from the relation

cos120593 =1198961+ 1198962

1198962minus 1198961

(34)

Under such circumstances in order to determine para-metric equations of the transition curve for 119897 isin (119897

0 119897119896⟩ func-

tions cosΘ(119897) and sinΘ(119897) should be expanded into Taylor

series after integration the following parametric equations ofthe transition curve are obtained

119909 (119897) = int cosΘ (119897) 119889119897

= 119909 (1198970) + cosΘ(119897

0) (119897 minus 119897

0)

minus1

6sinΘ(119897

0)Θ10158401015840(1198970) (119897 minus 119897

0)3

minus1

24sinΘ(119897

0)Θ101584010158401015840

(1198970) (119897 minus 119897

0)4

minus 1

40cosΘ(119897

0) [Θ10158401015840(1198970)]2

+1

120sinΘ(119897

0)Θ(4)

(1198970) (119897 minus 119897

0)5

minus1

72cosΘ(119897

0)Θ10158401015840(1198970)Θ101584010158401015840

(1198970) (119897 minus 119897

0)6

+ 1

336sinΘ(119897

0) [Θ10158401015840(1198970)]3

minus1

336cosΘ(119897

0)Θ10158401015840(1198970)Θ(4)

(1198970)

minus1

504cosΘ(119897

0) [Θ101584010158401015840

(1198970)]2

(119897 minus 1198970)7

+ 1

384sinΘ(119897

0) [Θ10158401015840(1198970)]2

Θ101584010158401015840

(1198970)

minus1

1152cosΘ(119897

0)Θ101584010158401015840

(1198970)Θ(4)

(1198970) (119897 minus 119897

0)8

+ 1

3456cosΘ(119897

0) [Θ10158401015840(1198970)]4

+1

1296sinΘ(119897

0)Θ10158401015840(1198970) [Θ101584010158401015840

(1198970)]2

+1

1728sinΘ(119897

0) [Θ10158401015840(1198970)]2

Θ(4)

(1198970)

minus1

10368cosΘ(119897

0) [Θ(4)

(1198970)]2

(119897 minus 1198970)9+ sdot sdot sdot

119910 (119897) = int sinΘ (119897) 119889119897

= 119910 (1198970) + sinΘ(119897

0) (119897 minus 119897

0)

+1

6[cosΘ(119897

0)Θ10158401015840(1198970)] (119897 minus 119897

0)3

+1

24[cosΘ(119897

0)Θ101584010158401015840

(1198970)] (119897 minus 119897

0)4

minus 1

40sinΘ(119897

0) [Θ10158401015840(1198970)]2

minus1

120cosΘ(119897

0)Θ(4)

(1198970) (119897 minus 119897

0)5

minus1

72[sinΘ(119897

0)Θ10158401015840(1198970)Θ101584010158401015840

(1198970)] (119897 minus 119897

0)6

Mathematical Problems in Engineering 7

minus 1

336cosΘ(119897

0) [Θ10158401015840(1198970)]3

+1

336sinΘ(119897

0)Θ10158401015840(1198970)Θ(4)

(1198970)

+1

504sinΘ(119897

0) [Θ101584010158401015840

(1198970)]2

(119897 minus 1198970)7

minus 1

384cosΘ(119897

0) [Θ10158401015840(1198970)]2

Θ101584010158401015840

(1198970)

+1

1152sinΘ(119897

0)Θ101584010158401015840

(1198970)Θ(4)

(1198970) (119897 minus 119897

0)8

+ 1

3456sinΘ(119897

0) [Θ10158401015840(1198970)]4

minus1

1296cosΘ(119897

0)Θ10158401015840(1198970) [Θ101584010158401015840

(1198970)]2

minus1

1728cosΘ(119897

0) [Θ10158401015840(1198970)]2

Θ(4)

(1198970)

minus1

10368sinΘ(119897

0) [Θ(4)

(1198970)]2

(119897 minus 1198970)9+ sdot sdot sdot

(35)

where

Θ(1198970) = 11989611198970+

1

1198972

119896

(1198962minus 1198961) 1198973

0minus

1

21198973

119896

(1198962minus 1198961) 1198974

0

Θ10158401015840(1198970) =

6

1198972

119896

(1198962minus 1198961) 1198970minus

6

1198973

119896

(1198962minus 1198961) 1198972

0

Θ101584010158401015840

(1198970) =

6

1198972

119896

(1198962minus 1198961) minus

12

1198973

119896

(1198962minus 1198961) 1198970

Θ(4)

(1198970) = minus

12

1198973

119896

(1198962minus 1198961)

(36)

Making use of parametric equations (31) and also (35) onecan obtain a correct solution of the problem with diversifiedsigns of curvature

Value of the tangent slope at the end of the transitioncurvemakes it possible to connect to it a curve of curvature 119896

2

satisfying simultaneously the tangency condition of both thecurves The above value is the same as in the case of linearcurvature It is defined by (25)

5 Transition Curve of Curvature inTrigonometric Form

51 Formulation of Curvature Equation As before boundaryconditions (26) and another differential equation areassumed

119896(4)

(119897) +1205872

1198972

119896

11989610158401015840(119897) = 0 (37)

As soon as the differential problem (26) (37) has beensolved a formula for curvature is obtained

119896 (119897) =1

2(1198961+ 1198962) minus

1

2(1198962minus 1198961) cos 120587

119897119896

119897 (38)

The form of function Θ(119897) determined by the use of (5)and (38) is as follows

Θ (119897) =1

2(1198961+ 1198962) 119897 minus

1

2(1198962minus 1198961)119897119896

120587sin 120587

119897119896

119897 (39)

The length of the transition curve on railway routesfollows from conditions

119897120595

119896ge

120587

2

V120595dop

10038161003816100381610038161198862 minus 1198861

1003816100381610038161003816

119897119891

119896ge

120587

2

V119891dop

1003816100381610038161003816ℎ2 minus ℎ1

1003816100381610038161003816

(40)

The assumed length of the curve must satisfy the condi-tion (19)

52 Ordinates of Transition Curve Connecting UniformCurva-tures After the expansion of functions cosΘ(119897) and sinΘ(119897)

intoMaclaurin series to be next integrated the following para-metric equations are obtained

119909 (119897) = int cosΘ (119897) 119889119897

= 119897 minus1198962

1

61198973+ [

1198964

1

120minus

1

60

1205872

1198972

119896

1198961(1198962minus 1198961)] 1198975

minus [1198966

1

5040minus

1

504

1205872

1198972

119896

1198963

1(1198962minus 1198961)

minus1

1680

1205874

1198974

119896

1198961(1198962minus 1198961) minus

1

2016

1205874

1198974

119896

(1198962minus 1198961)2] 1198977

+ [1198968

1

362880minus

1

12960

1205872

1198972

119896

1198965

1(1198962minus 1198961)

minus1

5184

1205874

1198974

119896

1198962

1(1198962minus 1198961)2

minus1

12960

1205874

1198974

119896

1198963

1(1198962minus 1198961) minus

1

90720

1205876

1198976

119896

1198961(1198962minus 1198961)

+1

25920

1205876

1198976

119896

(1198962minus 1198961)2] 1198979+ sdot sdot sdot

8 Mathematical Problems in Engineering

119910 (119897) = int sinΘ (119897) 119889119897

=1

211989611198972minus [

1

241198963

1minus

1

48

1205872

1198972

119896

(1198962minus 1198961)] 1198974

+ [1

7201198965

1minus

1

1441198962

1

1205872

1198972

119896

(1198962minus 1198961)

minus1

1440

1205874

1198974

119896

(1198962minus 1198961)] 1198976

minus [1

403201198967

1minus

1

23041198964

1

1205872

1198972

119896

(1198962minus 1198961)

+1

2304

1205874

1198974

119896

1198961(1198962minus 1198961)2minus

1

3840

1205874

1198974

119896

1198962

1(1198962minus 1198961)

+1

80640

1205876

1198976

119896

(1198962minus 1198961)] 1198978+ sdot sdot sdot

(41)

Equations (41) for 1198961

= 0 describe the curve known ascosinusoid which can be used to connect a straight with acircular arc

53 Transition Curve Coordinates Connecting Inverse Curva-tures Near inverse arcs in the diagram of functionΘ(119897) thereappears the extremum at point 119897

0 where

Θ1015840(1198970) = 119896 (119897

0) =

1

2(1198961+ 1198962) minus

1

2(1198962minus 1198961) cos 120587

119897119896

1198970= 0

(42)

The transition curve parametric equations (41) are effec-tive for 119897 isin ⟨0 119897

0⟩ To determine value 119897

0one must solve (42)

The equation can be written in the form

cos 120587

119897119896

1198970=

1198961+ 1198962

1198962minus 1198961

(43)

It is easy to prove that for sign 1198961

= sign 1198962it is necessary

to satisfy the condition |1198961+1198962| lt |1198962minus1198961| thus |(119896

1+1198962)(1198962minus

1198961)| lt 1 The formula for length 119897

0is as follows

1198970=

1

120587(arccos1198961 + 119896

2

1198962minus 1198961

) 119897119896 (44)

For 119897 isin (1198970 119897119896⟩ functions cosΘ(119897) and sinΘ(119897) should

be expanded into Taylor series The integration provides thefollowing equations

119909 (119897) = int cosΘ (119897) 119889119897

= 119909 (1198970) + cosΘ(119897

0) (119897 minus 119897

0)

minus1

6sinΘ(119897

0)Θ10158401015840(1198970) (119897 minus 119897

0)3

minus1

24sinΘ(119897

0)Θ101584010158401015840

(1198970) sdot (119897 minus 119897

0)4

minus 1

40cosΘ(119897

0) [Θ10158401015840(1198970)]2

+1

120sinΘ(119897

0)Θ(4)

(1198970) (119897 minus 119897

0)5

minus 1

72cosΘ(119897

0)Θ10158401015840(1198970)Θ101584010158401015840

(1198970)

+1

720sinΘ(119897

0)Θ(5)

(1198970) (119897 minus 119897

0)6

+ 1

336sinΘ(119897

0) [Θ10158401015840(1198970)]3

minus1

336cosΘ(119897

0)Θ10158401015840(1198970)Θ(4)

(1198970)

minus1

504cosΘ(119897

0) [Θ101584010158401015840

(1198970)]2

minus1

5040sinΘ(119897

0)Θ(6)

(1198970) (119897 minus 119897

0)7

+ 1

384sinΘ(119897

0) [Θ10158401015840(1198970)]2

Θ101584010158401015840

(1198970)

minus1

1920cosΘ(119897

0)Θ10158401015840(119897) Θ(5)

(119897)

minus1

1152cosΘ(119897

0)Θ101584010158401015840

(1198970)Θ(4)

(1198970)

minus1

40320sinΘ(119897

0)Θ(7)

(1198970) (119897 minus 119897

0)8

+ 1

3456cosΘ(119897

0) [Θ10158401015840(1198970)]4

+1

1296sinΘ(119897

0)Θ10158401015840(1198970) [Θ101584010158401015840

(1198970)]2

+1

1728sinΘ(119897

0) [Θ10158401015840(1198970)]2

Θ(4)

(1198970)

minus1

12960cosΘ(119897

0)Θ10158401015840(1198970)Θ(6)

(1198970)

minus1

6480cosΘ(119897

0)Θ101584010158401015840

(1198970)Θ(5)

(1198970)

minus1

10368cosΘ(119897

0) [Θ(4)

(1198970)]2

minus1

362880sinΘ(119897

0)Θ(8)

(1198970) (119897 minus 119897

0)9+ sdot sdot sdot

Mathematical Problems in Engineering 9

119910 (119897) = int sinΘ (119897) 119889119897

= 119910 (1198970) + sinΘ(119897

0) (119897 minus 119897

0)

+1

6cosΘ(119897

0)Θ10158401015840(1198970) (119897 minus 119897

0)3

+1

24cosΘ(119897

0)Θ101584010158401015840

(1198970) sdot (119897 minus 119897

0)4

minus 1

40sinΘ(119897

0) [Θ10158401015840(1198970)]2

minus1

120cosΘ(119897

0)Θ(4)

(1198970) (119897 minus 119897

0)5

minus 1

72sinΘ(119897

0) 11987610158401015840(1198970)Θ101584010158401015840

(1198970)

minus1

720cosΘ(119897

0)Θ(5)

(1198970) (119897 minus 119897

0)6

minus 1

336cosΘ(119897

0) [Θ10158401015840(1198970)]3

+1

336sinΘ(119897

0)Θ10158401015840(1198970)Θ(4)

(1198970)

+1

504sinΘ(119897

0) [Θ101584010158401015840

(1198970)]2

minus1

5040cosΘ(119897

0)Θ(6)

(1198970) (119897 minus 119897

0)7

minus 1

384cosΘ(119897

0) [Θ10158401015840(1198970)]2

Θ101584010158401015840

(1198970)

+1

1920sinΘ(119897

0)Θ10158401015840(1198970)Θ(5)

(1198970)

+1

1152sinΘ(119897

0)Θ101584010158401015840

(1198970)Θ(4)

(1198970)

minus1

40320cosΘ(119897

0)Θ(7)

(1198970) (119897 minus 119897

0)8

+ 1

3456sinΘ(119897

0) [Θ10158401015840(1198970)]4

minus1

1296cosΘ(119897

0)Θ10158401015840(1198970) [Θ101584010158401015840

(1198970)]2

minus1

1728cosΘ(119897

0) [Θ10158401015840(1198970)]2

Θ(4)

(1198970)

minus1

12960sin119876 (119897

0) 11987610158401015840(1198970) 119876(6)

(1198970)

minus1

6480sin119876 (119897

0) 119876101584010158401015840

(1198970) 119876(5)

(1198970)

minus1

10368sinΘ(119897

0) [Θ(4)

(1198970)]2

+1

362880cosΘ(119897

0)Θ(8)

(1198970) (119897 minus 119897

0)9+ sdot sdot sdot

(45)

where

Θ(1198970) =

1

2(1198961+ 1198962) 1198970plusmn

119897119896

120587radicminus11989611198962

Θ10158401015840(1198970) = ∓

120587

119897119896

radicminus11989611198962

Θ10158401015840(1198970) = ∓

120587

119897119896

radicminus11989611198962

Θ101584010158401015840

(1198970) =

1

2

1205872

1198972

119896

(1198961+ 1198962)

Θ(4)

(1198970) = plusmn

1205873

1198973

119896

radicminus11989611198962

Θ(5)

(1198970) = minus

1

2

1205874

1198974

119896

(1198961+ 1198962)

Θ(6)

(1198970) = ∓

1205875

1198975

119896

radicminus11989611198962

Θ(7)

(1198970) =

1

2

1205876

1198976

119896

(1198961+ 1198962)

Θ(8)

(1198970) = plusmn

1205877

1198977

119896

radicminus11989611198962

(46)

In expressions with double signs plusmn and ∓ the upper onecorresponds to the case 119896

1gt 0 119896

2lt 0 while the lower one

refers to 1198961lt 0 119896

2gt 0

The value of tangent slope at the end of the transitioncurvemakes it possible to connect to it a curve of curvature 119896

2

satisfying the tangency condition of both the curves Thevalue is the same as in the previous cases It is determinedby (25)

6 Comparison of Horizontal Ordinates ofDetermined Transition Curves

61 The Shaping of Horizontal Ordinates of ComparableCurves If the analysis is carried out for railway routes thenthe necessity of meeting kinematic conditions (10) and (11)will require a diversification of the lengths of the comparedwith each other curves (using the same speed of trains)Moreover it is necessary to individually consider cases ofconnecting homogeneous curvatures (sign 119896

1= sign 119896

2) and

inverse arcs (sign 1198961

= sign 1198962)

It is assumed that at joints of consistent arcs with radii1198771= 1200m and 119877

2= 700m an attempt is made to attain a

speed V = 110 kmh Hence it is necessary to employ on arcssuperelevation of ℎ

1= 80mmand ℎ

2= 115mm respectively

The retention of the same kinematic parameters can providediversified lengths of the transition curve

(i) for a curve of linear curvature 119897119896= 40m

(ii) for a curve of polynomial curvature 119897119896= 60m

(iii) for a curve of trigonometric curvature 119897119896= 62 832m

10 Mathematical Problems in Engineering

The above length relations result directly from the equa-tions given at Sections 32 41 and 51

Figure 5 gives diagrams of horizontal ordinates for com-pared transition curves of positive and negative curvaturevalues As seen the curvature ordinates of linear curvatureevidently deviate from the ordinates of other curves In turncurves of polynomial and trigonometric curvature are inprinciple consistent A difference can only be noted in thefinal area And so the class of the function describing thecurvature plays here a significant part with respect to a curveof linear curvature it is a function of class C0 while regardingthe other two curves it is a function of class C1

At the joint of inverse arcs of radii 1198771= 1200m and 119877

2=

700m an attempt is made to reach the speed of V = 90 kmhHence it follows that it is necessary to apply superelevationalong arcs attaining ℎ

1= 20mm and ℎ

2= 45mm respec-

tively By the use the same kinematic parameters it is possibleto obtain the following lengths of the transition curve

(i) for a curve of linear curvature 119897119896= 60m

(ii) for a curve of polynomial curvature 119897119896= 90m

(iii) for a curve of trigonometric curvature 119897119896= 94 248m

Figure 6 presents diagrams of horizontal ordinates ofcompared transition curves for positive and negative valuesof curvature

As can be seen like in the case of consistent arcs the curveordinates of linear curvature deviate significantly from theordinates of the remaining curves but the shape of thesecurves also differs one from the other The class of functiondescribing the curvature is still important However in therange of the same class various geometric solutions arepossible

62 Negotiation of Transition Curves in a Geometric SystemThe transition curves do not appear alone in a geometricsystem but must be connected with some arcs in the neigh-borhood If the input arc is circular then its equation is asfollows

(i) for 1198961gt 0

119910 (119909) = 1198771minus radic1198772

1minus 1199092 (47)

(ii) for 1198961lt 0

119910 (119909) = minus (1198771minus radic1198772

1minus 1199092) (48)

where 119909 lt 0

However if the output arc is also a circular arc then it isexpressed in the following equation

(i) for 1198962gt 0

119910 (119909) = 119910119878minus radic1198772

2minus (119909119878minus 119909)2

119909119878= 119909119870

minus119904119870

radic1 + 1199042

119870

1198772 119910

119878= 119910119870

+1

radic1 + 1199042

119870

1198772

(49)

0051

152

25

0 10 20 30 40 50 60 70x (m)

y(m

)

minus05

minus1

minus15

minus2

minus25

Figure 5 Diagrams of horizontal ordinates of curves joiningconsistent arcs for positive and negative values of curvature (usingdifferent horizontal and vertical scales) curve of linear curvatureline in violet colour curve of polynomial curvature in green colourand curve of trigonometric curvature in red colour

0

05

1

15

0 20 40 60 80 100x (m)

y(m

)

minus05

minus1

minus15

Figure 6 Diagrams of horizontal ordinates of curves connectinginverse arcs for positive and negative values of curvature (usingdifferent horizontal and vertical scales) the curve of linear curvatureis in violet color the curve of polynomial curvature is in green colorand the curve of trigonometric curvature is in red color

(ii) for 1198962lt 0

119910 (119909) = 119910119878+ radic1198772

2minus (119909119878minus 119909)2

119909119878= 119909119870

+119904119870

radic1 + 1199042

119870

1198772 119910

119878= 119910119870

minus1

radic1 + 1199042

119870

1198772

(50)

where 119909 gt 119909119870

In (49) and (50) 119909119870denotes the abscissae of the transition

curve end119910119870is the ordinate of the transition curve endwhile

119904119870is the value of tangent at the end of the transition curve

Values for numerical examples under consideration are givenin Table 1

Figures 7 and 8 illustrate the geometric systems made upof two circular arcs of consistent curvature connected withvarious types of transition curves As seen the application of agiven type of transition curve leads to an adequate position ofthe output circular arcThe position of this arc by the use of alinear transition curve clearly deviates from such cases whereadvantage is taken of curves with polynomial and trigono-metrical curvatureWith regard to the latter amutual positionof the circular arc varies insignificantly

Mathematical Problems in Engineering 11

Table 1 Comparison of values 119909119870 119910119870 and 119904

119870for numerical examples analyzed

Curve 119897119896[m] Θ(119897

119896) [rad] 119904

119870119909119870[m] 119910

119870[m]

Case 1 1198961= 11200 radm and 119896

2= 1700 radm

Curve C0 40 0045238 0045269 39988 0825Curve C1 (p) 60 0067857 0067961 59963 1821Curve C1 (t) 62832 0071060 0071180 62796 1940

Case 2 1198961= 11200 radm and 119896

2= minus1700 radm

Curve C0 60 minus0017857 minus0017859 59999 0143Curve C1 (p) 90 minus0026786 minus0026792 89991 0627Curve C1 (t) 94248 minus0028050 minus0028057 94238 0714

Case 3 1198961= minus11200 radm and 119896

2= 1700 radm

Curve C0 60 001786 0017859 59999 minus0143Curve C1 (p) 90 0026786 0026792 89991 minus0627Curve C1 (t) 94248 0028050 0028057 94238 minus0714

Case 4 1198961= minus11200 radm and 119896

2= minus1700 radm

Curve C0 40 minus0045238 minus0045269 39988 minus0825Curve C1 (p) 60 minus0067857 minus0067961 59969 minus1821Curve C1 (t) 62832 minus0071060 minus0071180 62976 minus1940

0

1

2

3

4

5

6

0 20 40 60 80 100 120x (m)

y(m

)

minus40 minus20

Figure 7 Examples of geometric systems consisting of two circulararcs with consistent curvature (brown line) connected with differenttransition curves for positive values of curvature (using differenthorizontal and vertical scales) curve of linear curvature in violetline curve of polynomial curvature in green and curve of trigono-metric curvature in red

00 20 40 60 80 100 120

x (m)

y(m

)

minus40 minus20

minus1

minus2

minus3

minus4

minus5

minus6

minus7

Figure 8 Examples of geometric systems consisting of two circulararcs with consistent curvature (brown line) connected with differenttransition curves for negative values of curvature (using differenthorizontal and vertical scales) curve of linear curvature in violetline curve of polynomial curvature in green and curve of trigono-metric curvature in red

0

05

1

15

0 50 100 150x (m)

y(m

)

minus05

minus1

minus15

minus2

minus50

Figure 9 Examples of geometric systems consisting of two inversecircular arcs (brown line) connected with different types of transi-tion curves for positive values of input curvature (using differenthorizontal and vertical scales) curve of linear curvature in violetline curve of polynomial curvature in green and curve of trigono-metric curvature in red

Figures 9 and 10 illustrate geometric systems made up oftwo inverse circular arcs connected with various types oftransition curves

Here the layout of the output circular arc is much morediversifiedThe position of the arc obtained by the applicationof the transition curve of linear curvature diverts by severaldozenmeters from situationswhere curves of polynomial andtrigonometric curvature have been used The difference forcurves of polynomial and trigonometric curvature attains inthe numerical example analyzed an order of several meters

The presented examples are taken in the very nature ofthings at random They reveal wide possibilities for the pre-sented analytical method in modelling geometric systems Itsapplication makes it possible to generate diversified layout ofthe communication routes adapted to the field conditions andother circumstances Simultaneously the method ensurescorrectness of the obtained solution in view of the kinematicparameters magnitudes in force

12 Mathematical Problems in Engineering

0

05

1

15

2

0 50 100 150x (m)

y(m

)

minus05

minus1

minus15

minus50

Figure 10 Examples of geometric systems consisting of twoinverse circular arcs (brown line) connected with different typesof transition curves for negative values of input curvature (usingdifferent horizontal and vertical scales) curve of linear curvaturein violet line curve of polynomial curvature in green and curve oftrigonometric curvature in red

7 Conclusions

The presented universal method of modelling geometricalsystem of a communication route is based on the determina-tion of an adequate curvature equation by using differentialequationsThe technique enables us to design connections ofvarious geometrical elements of the route indicating diversifi-cation in its curvature It is characterized by an analytical wayof recording certain functions and moreover it is possible tospecify in advance the class of these functions

The sought-after connection is recorded in the form ofparametric equations 119909(119897) and 119910(119897) where parameter 119897 isthe position of a given point along the length of the curveThe determination of 119909(119897) and 119910(119897) needs the expansion ofintegrand functions into Maclaurin series and with regard toinverse curvatures into Taylor series

The determination of coordinates of an adequate tran-sition curve in rectangular coordinate system 119909 119910 makesit possible for an easy transfer of it to Gauss-Kruger (119883 119884)

conformal coordinates and its subsequent layout in fieldmaking use of the satellite measuring technique GNSS

The method can be applied to identify various typesof transition curves The paper gives examples of solvingthe problem involving linear change of curvature and thecurvature in polynomial and trigonometric form

The performed comparative analysis reveals a wide rangeof possibilities which are offered by the presented designmethod Its application enables us to generate diversifiedlayout of the route It simultaneously ensures a completecontrol of correctness of the obtained solutions in view ofsatisfying the geometric and kinematic conditions

The presented method of modelling curvature has auniversal character It can be used for vehicular roads andrailway lines as well In the case of railway lines therearises an additional possibility of modelling in a similar waythe superelevation ramp understood here as a determineddifference in the height of the rail courses

Conflict of Interests

The author declares that he has no conflict of interestsregarding the publication of this paper

References

[1] W Koc and C Specht ldquoApplication of the Polish active GNSSgeodetic network for surveying and design of the railroadrdquo inProceedings of the 1st International Conference on Road and RailInfrastructure (CETRA rsquo10) pp 757ndash762 May 2010

[2] P Lipar M Lakner T Maher andM Zura ldquoEstimation of roadcenterline curvature from raw GPS datardquo Baltic Journal of Roadand Bridge Engineering vol 6 no 3 pp 163ndash168 2011

[3] L Tasci and N Kuloglu ldquoInvestigation of a new transitioncurverdquoTheBaltic Journal of Road and Bridge Engineering vol 6no 1 pp 23ndash29 2011

[4] H-H Cai and G-J Wang ldquoA new method in highway routedesign joining circular arcs by a single C-Bezier curve withshape parameterrdquo Journal of Zhejiang University Science A vol10 no 4 pp 562ndash569 2009

[5] S Dimulyo Z Habib and M Sakai ldquoFair cubic transition bet-ween two circles with one circle inside or tangent to the otherrdquoNumerical Algorithms vol 51 no 4 pp 461ndash476 2009

[6] G Farin ldquoClass A Bezier curvesrdquo Computer Aided GeometricDesign vol 23 no 7 pp 573ndash581 2006

[7] Z Habib and M Sakai ldquoG2 pythagorean hodograph quintictransition between two circles with shape controlrdquo ComputerAided Geometric Design vol 24 no 5 pp 252ndash266 2007

[8] G Harary and A Tal ldquoThe natural 3D spiralrdquo Computer Graph-ics Forum vol 30 no 2 pp 237ndash246 2011

[9] K T Miura ldquoA general equation of aesthetic curves and its self-affinityrdquo Computer-Aided Design and Applications vol 3 no 1ndash4 pp 457ndash464 2006

[10] S H Yahaya M S Salleh and J M Ali ldquoSpur gear design withan S-shaped transition curve application using mathematicaand CAD toolsrdquo in Proceedings of the International Conferenceon Computer Technology and Development (ICCTD rsquo09) pp273ndash276 Dubai United Arab Emirates November 2009

[11] N Yoshida and T Saito ldquoQuasi-aesthetic curves in rationalcubic Bezier formsrdquo Computer-Aided Design and Applicationsvol 4 no 1ndash6 pp 477ndash486 2007

[12] R Ziatdinov N Yoshida and T-W Kim ldquoAnalytic parametricequations of log-aesthetic curves in terms of incomplete gammafunctionsrdquo Computer Aided Geometric Design vol 29 no 2 pp129ndash140 2012

[13] R J Grabowski ldquoSmooth curvilinear transitions along vehicu-lar roads and railway linesrdquo Science Papers of AGH Universityof Science and Technology no 82 Cracow Poland 1984(Polish)

[14] L T Klauder ldquoTrack transition curve geometry based on gegen-bauer polynomialsrdquo in Proceedings of the 6th International Con-ference on Railway Engineering May 2003

[15] W Koc Elements of Design Theory on Track Systems GdanskUniversity of Technology Gdansk Poland 2004 (Polish)

[16] E Mieloszyk and W Koc ldquoGeneral dynamic method for det-ermining transition curve equationsrdquo Rail InternationalmdashSchienen der Welt vol 10 pp 32ndash40 1991

[17] B Kufver ldquoRealigning railway in track renewalsmdashlinear versusS-shaped superelevation rampsrdquo in Proceedings of the 2ndInternational Conference on Railway Engineering May 1999

[18] W Koc and E Mieloszyk ldquoComparative analysis of selectedtransition curves by the use of dynamicmodelrdquoArchives of CivilEngineering vol 33 no 2 pp 239ndash261 1987 (Polish)

[19] S Gleason and D Gebre-Egziabher GNSS Applications andMethods Artech House Publishers Norwood NJ USA 2009

Mathematical Problems in Engineering 13

[20] B Hoffmann-Wellenhof H Lichtenegger and E Wasle GNSSGlobal Navigation Satellite SystemsmdashGPS GLONASS Galileo ampMore Springer Wien Austria 2008

[21] P Misra and P Enge Global Positioning System Signals Mea-surements and Performance Ganga-Jamuna Press LincolnMass USA 2012

[22] C Specht ldquoAccuracy and coverage of the modernized PolishMaritime differential GPS systemrdquo Advances in Space Researchvol 47 no 2 pp 221ndash228 2011

[23] W Koc and K Palikowska ldquoDynamic properties evaluation ofthe selected methods of joining route segments with differentcurvaturerdquo Technika Transportu Szynowego vol 9 pp 1785ndash1807 2012 (Polish)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Complex AnalysisJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

minus 1

336cosΘ(119897

0) [Θ10158401015840(1198970)]3

+1

336sinΘ(119897

0)Θ10158401015840(1198970)Θ(4)

(1198970)

+1

504sinΘ(119897

0) [Θ101584010158401015840

(1198970)]2

(119897 minus 1198970)7

minus 1

384cosΘ(119897

0) [Θ10158401015840(1198970)]2

Θ101584010158401015840

(1198970)

+1

1152sinΘ(119897

0)Θ101584010158401015840

(1198970)Θ(4)

(1198970) (119897 minus 119897

0)8

+ 1

3456sinΘ(119897

0) [Θ10158401015840(1198970)]4

minus1

1296cosΘ(119897

0)Θ10158401015840(1198970) [Θ101584010158401015840

(1198970)]2

minus1

1728cosΘ(119897

0) [Θ10158401015840(1198970)]2

Θ(4)

(1198970)

minus1

10368sinΘ(119897

0) [Θ(4)

(1198970)]2

(119897 minus 1198970)9+ sdot sdot sdot

(35)

where

Θ(1198970) = 11989611198970+

1

1198972

119896

(1198962minus 1198961) 1198973

0minus

1

21198973

119896

(1198962minus 1198961) 1198974

0

Θ10158401015840(1198970) =

6

1198972

119896

(1198962minus 1198961) 1198970minus

6

1198973

119896

(1198962minus 1198961) 1198972

0

Θ101584010158401015840

(1198970) =

6

1198972

119896

(1198962minus 1198961) minus

12

1198973

119896

(1198962minus 1198961) 1198970

Θ(4)

(1198970) = minus

12

1198973

119896

(1198962minus 1198961)

(36)

Making use of parametric equations (31) and also (35) onecan obtain a correct solution of the problem with diversifiedsigns of curvature

Value of the tangent slope at the end of the transitioncurvemakes it possible to connect to it a curve of curvature 119896

2

satisfying simultaneously the tangency condition of both thecurves The above value is the same as in the case of linearcurvature It is defined by (25)

5 Transition Curve of Curvature inTrigonometric Form

51 Formulation of Curvature Equation As before boundaryconditions (26) and another differential equation areassumed

119896(4)

(119897) +1205872

1198972

119896

11989610158401015840(119897) = 0 (37)

As soon as the differential problem (26) (37) has beensolved a formula for curvature is obtained

119896 (119897) =1

2(1198961+ 1198962) minus

1

2(1198962minus 1198961) cos 120587

119897119896

119897 (38)

The form of function Θ(119897) determined by the use of (5)and (38) is as follows

Θ (119897) =1

2(1198961+ 1198962) 119897 minus

1

2(1198962minus 1198961)119897119896

120587sin 120587

119897119896

119897 (39)

The length of the transition curve on railway routesfollows from conditions

119897120595

119896ge

120587

2

V120595dop

10038161003816100381610038161198862 minus 1198861

1003816100381610038161003816

119897119891

119896ge

120587

2

V119891dop

1003816100381610038161003816ℎ2 minus ℎ1

1003816100381610038161003816

(40)

The assumed length of the curve must satisfy the condi-tion (19)

52 Ordinates of Transition Curve Connecting UniformCurva-tures After the expansion of functions cosΘ(119897) and sinΘ(119897)

intoMaclaurin series to be next integrated the following para-metric equations are obtained

119909 (119897) = int cosΘ (119897) 119889119897

= 119897 minus1198962

1

61198973+ [

1198964

1

120minus

1

60

1205872

1198972

119896

1198961(1198962minus 1198961)] 1198975

minus [1198966

1

5040minus

1

504

1205872

1198972

119896

1198963

1(1198962minus 1198961)

minus1

1680

1205874

1198974

119896

1198961(1198962minus 1198961) minus

1

2016

1205874

1198974

119896

(1198962minus 1198961)2] 1198977

+ [1198968

1

362880minus

1

12960

1205872

1198972

119896

1198965

1(1198962minus 1198961)

minus1

5184

1205874

1198974

119896

1198962

1(1198962minus 1198961)2

minus1

12960

1205874

1198974

119896

1198963

1(1198962minus 1198961) minus

1

90720

1205876

1198976

119896

1198961(1198962minus 1198961)

+1

25920

1205876

1198976

119896

(1198962minus 1198961)2] 1198979+ sdot sdot sdot

8 Mathematical Problems in Engineering

119910 (119897) = int sinΘ (119897) 119889119897

=1

211989611198972minus [

1

241198963

1minus

1

48

1205872

1198972

119896

(1198962minus 1198961)] 1198974

+ [1

7201198965

1minus

1

1441198962

1

1205872

1198972

119896

(1198962minus 1198961)

minus1

1440

1205874

1198974

119896

(1198962minus 1198961)] 1198976

minus [1

403201198967

1minus

1

23041198964

1

1205872

1198972

119896

(1198962minus 1198961)

+1

2304

1205874

1198974

119896

1198961(1198962minus 1198961)2minus

1

3840

1205874

1198974

119896

1198962

1(1198962minus 1198961)

+1

80640

1205876

1198976

119896

(1198962minus 1198961)] 1198978+ sdot sdot sdot

(41)

Equations (41) for 1198961

= 0 describe the curve known ascosinusoid which can be used to connect a straight with acircular arc

53 Transition Curve Coordinates Connecting Inverse Curva-tures Near inverse arcs in the diagram of functionΘ(119897) thereappears the extremum at point 119897

0 where

Θ1015840(1198970) = 119896 (119897

0) =

1

2(1198961+ 1198962) minus

1

2(1198962minus 1198961) cos 120587

119897119896

1198970= 0

(42)

The transition curve parametric equations (41) are effec-tive for 119897 isin ⟨0 119897

0⟩ To determine value 119897

0one must solve (42)

The equation can be written in the form

cos 120587

119897119896

1198970=

1198961+ 1198962

1198962minus 1198961

(43)

It is easy to prove that for sign 1198961

= sign 1198962it is necessary

to satisfy the condition |1198961+1198962| lt |1198962minus1198961| thus |(119896

1+1198962)(1198962minus

1198961)| lt 1 The formula for length 119897

0is as follows

1198970=

1

120587(arccos1198961 + 119896

2

1198962minus 1198961

) 119897119896 (44)

For 119897 isin (1198970 119897119896⟩ functions cosΘ(119897) and sinΘ(119897) should

be expanded into Taylor series The integration provides thefollowing equations

119909 (119897) = int cosΘ (119897) 119889119897

= 119909 (1198970) + cosΘ(119897

0) (119897 minus 119897

0)

minus1

6sinΘ(119897

0)Θ10158401015840(1198970) (119897 minus 119897

0)3

minus1

24sinΘ(119897

0)Θ101584010158401015840

(1198970) sdot (119897 minus 119897

0)4

minus 1

40cosΘ(119897

0) [Θ10158401015840(1198970)]2

+1

120sinΘ(119897

0)Θ(4)

(1198970) (119897 minus 119897

0)5

minus 1

72cosΘ(119897

0)Θ10158401015840(1198970)Θ101584010158401015840

(1198970)

+1

720sinΘ(119897

0)Θ(5)

(1198970) (119897 minus 119897

0)6

+ 1

336sinΘ(119897

0) [Θ10158401015840(1198970)]3

minus1

336cosΘ(119897

0)Θ10158401015840(1198970)Θ(4)

(1198970)

minus1

504cosΘ(119897

0) [Θ101584010158401015840

(1198970)]2

minus1

5040sinΘ(119897

0)Θ(6)

(1198970) (119897 minus 119897

0)7

+ 1

384sinΘ(119897

0) [Θ10158401015840(1198970)]2

Θ101584010158401015840

(1198970)

minus1

1920cosΘ(119897

0)Θ10158401015840(119897) Θ(5)

(119897)

minus1

1152cosΘ(119897

0)Θ101584010158401015840

(1198970)Θ(4)

(1198970)

minus1

40320sinΘ(119897

0)Θ(7)

(1198970) (119897 minus 119897

0)8

+ 1

3456cosΘ(119897

0) [Θ10158401015840(1198970)]4

+1

1296sinΘ(119897

0)Θ10158401015840(1198970) [Θ101584010158401015840

(1198970)]2

+1

1728sinΘ(119897

0) [Θ10158401015840(1198970)]2

Θ(4)

(1198970)

minus1

12960cosΘ(119897

0)Θ10158401015840(1198970)Θ(6)

(1198970)

minus1

6480cosΘ(119897

0)Θ101584010158401015840

(1198970)Θ(5)

(1198970)

minus1

10368cosΘ(119897

0) [Θ(4)

(1198970)]2

minus1

362880sinΘ(119897

0)Θ(8)

(1198970) (119897 minus 119897

0)9+ sdot sdot sdot

Mathematical Problems in Engineering 9

119910 (119897) = int sinΘ (119897) 119889119897

= 119910 (1198970) + sinΘ(119897

0) (119897 minus 119897

0)

+1

6cosΘ(119897

0)Θ10158401015840(1198970) (119897 minus 119897

0)3

+1

24cosΘ(119897

0)Θ101584010158401015840

(1198970) sdot (119897 minus 119897

0)4

minus 1

40sinΘ(119897

0) [Θ10158401015840(1198970)]2

minus1

120cosΘ(119897

0)Θ(4)

(1198970) (119897 minus 119897

0)5

minus 1

72sinΘ(119897

0) 11987610158401015840(1198970)Θ101584010158401015840

(1198970)

minus1

720cosΘ(119897

0)Θ(5)

(1198970) (119897 minus 119897

0)6

minus 1

336cosΘ(119897

0) [Θ10158401015840(1198970)]3

+1

336sinΘ(119897

0)Θ10158401015840(1198970)Θ(4)

(1198970)

+1

504sinΘ(119897

0) [Θ101584010158401015840

(1198970)]2

minus1

5040cosΘ(119897

0)Θ(6)

(1198970) (119897 minus 119897

0)7

minus 1

384cosΘ(119897

0) [Θ10158401015840(1198970)]2

Θ101584010158401015840

(1198970)

+1

1920sinΘ(119897

0)Θ10158401015840(1198970)Θ(5)

(1198970)

+1

1152sinΘ(119897

0)Θ101584010158401015840

(1198970)Θ(4)

(1198970)

minus1

40320cosΘ(119897

0)Θ(7)

(1198970) (119897 minus 119897

0)8

+ 1

3456sinΘ(119897

0) [Θ10158401015840(1198970)]4

minus1

1296cosΘ(119897

0)Θ10158401015840(1198970) [Θ101584010158401015840

(1198970)]2

minus1

1728cosΘ(119897

0) [Θ10158401015840(1198970)]2

Θ(4)

(1198970)

minus1

12960sin119876 (119897

0) 11987610158401015840(1198970) 119876(6)

(1198970)

minus1

6480sin119876 (119897

0) 119876101584010158401015840

(1198970) 119876(5)

(1198970)

minus1

10368sinΘ(119897

0) [Θ(4)

(1198970)]2

+1

362880cosΘ(119897

0)Θ(8)

(1198970) (119897 minus 119897

0)9+ sdot sdot sdot

(45)

where

Θ(1198970) =

1

2(1198961+ 1198962) 1198970plusmn

119897119896

120587radicminus11989611198962

Θ10158401015840(1198970) = ∓

120587

119897119896

radicminus11989611198962

Θ10158401015840(1198970) = ∓

120587

119897119896

radicminus11989611198962

Θ101584010158401015840

(1198970) =

1

2

1205872

1198972

119896

(1198961+ 1198962)

Θ(4)

(1198970) = plusmn

1205873

1198973

119896

radicminus11989611198962

Θ(5)

(1198970) = minus

1

2

1205874

1198974

119896

(1198961+ 1198962)

Θ(6)

(1198970) = ∓

1205875

1198975

119896

radicminus11989611198962

Θ(7)

(1198970) =

1

2

1205876

1198976

119896

(1198961+ 1198962)

Θ(8)

(1198970) = plusmn

1205877

1198977

119896

radicminus11989611198962

(46)

In expressions with double signs plusmn and ∓ the upper onecorresponds to the case 119896

1gt 0 119896

2lt 0 while the lower one

refers to 1198961lt 0 119896

2gt 0

The value of tangent slope at the end of the transitioncurvemakes it possible to connect to it a curve of curvature 119896

2

satisfying the tangency condition of both the curves Thevalue is the same as in the previous cases It is determinedby (25)

6 Comparison of Horizontal Ordinates ofDetermined Transition Curves

61 The Shaping of Horizontal Ordinates of ComparableCurves If the analysis is carried out for railway routes thenthe necessity of meeting kinematic conditions (10) and (11)will require a diversification of the lengths of the comparedwith each other curves (using the same speed of trains)Moreover it is necessary to individually consider cases ofconnecting homogeneous curvatures (sign 119896

1= sign 119896

2) and

inverse arcs (sign 1198961

= sign 1198962)

It is assumed that at joints of consistent arcs with radii1198771= 1200m and 119877

2= 700m an attempt is made to attain a

speed V = 110 kmh Hence it is necessary to employ on arcssuperelevation of ℎ

1= 80mmand ℎ

2= 115mm respectively

The retention of the same kinematic parameters can providediversified lengths of the transition curve

(i) for a curve of linear curvature 119897119896= 40m

(ii) for a curve of polynomial curvature 119897119896= 60m

(iii) for a curve of trigonometric curvature 119897119896= 62 832m

10 Mathematical Problems in Engineering

The above length relations result directly from the equa-tions given at Sections 32 41 and 51

Figure 5 gives diagrams of horizontal ordinates for com-pared transition curves of positive and negative curvaturevalues As seen the curvature ordinates of linear curvatureevidently deviate from the ordinates of other curves In turncurves of polynomial and trigonometric curvature are inprinciple consistent A difference can only be noted in thefinal area And so the class of the function describing thecurvature plays here a significant part with respect to a curveof linear curvature it is a function of class C0 while regardingthe other two curves it is a function of class C1

At the joint of inverse arcs of radii 1198771= 1200m and 119877

2=

700m an attempt is made to reach the speed of V = 90 kmhHence it follows that it is necessary to apply superelevationalong arcs attaining ℎ

1= 20mm and ℎ

2= 45mm respec-

tively By the use the same kinematic parameters it is possibleto obtain the following lengths of the transition curve

(i) for a curve of linear curvature 119897119896= 60m

(ii) for a curve of polynomial curvature 119897119896= 90m

(iii) for a curve of trigonometric curvature 119897119896= 94 248m

Figure 6 presents diagrams of horizontal ordinates ofcompared transition curves for positive and negative valuesof curvature

As can be seen like in the case of consistent arcs the curveordinates of linear curvature deviate significantly from theordinates of the remaining curves but the shape of thesecurves also differs one from the other The class of functiondescribing the curvature is still important However in therange of the same class various geometric solutions arepossible

62 Negotiation of Transition Curves in a Geometric SystemThe transition curves do not appear alone in a geometricsystem but must be connected with some arcs in the neigh-borhood If the input arc is circular then its equation is asfollows

(i) for 1198961gt 0

119910 (119909) = 1198771minus radic1198772

1minus 1199092 (47)

(ii) for 1198961lt 0

119910 (119909) = minus (1198771minus radic1198772

1minus 1199092) (48)

where 119909 lt 0

However if the output arc is also a circular arc then it isexpressed in the following equation

(i) for 1198962gt 0

119910 (119909) = 119910119878minus radic1198772

2minus (119909119878minus 119909)2

119909119878= 119909119870

minus119904119870

radic1 + 1199042

119870

1198772 119910

119878= 119910119870

+1

radic1 + 1199042

119870

1198772

(49)

0051

152

25

0 10 20 30 40 50 60 70x (m)

y(m

)

minus05

minus1

minus15

minus2

minus25

Figure 5 Diagrams of horizontal ordinates of curves joiningconsistent arcs for positive and negative values of curvature (usingdifferent horizontal and vertical scales) curve of linear curvatureline in violet colour curve of polynomial curvature in green colourand curve of trigonometric curvature in red colour

0

05

1

15

0 20 40 60 80 100x (m)

y(m

)

minus05

minus1

minus15

Figure 6 Diagrams of horizontal ordinates of curves connectinginverse arcs for positive and negative values of curvature (usingdifferent horizontal and vertical scales) the curve of linear curvatureis in violet color the curve of polynomial curvature is in green colorand the curve of trigonometric curvature is in red color

(ii) for 1198962lt 0

119910 (119909) = 119910119878+ radic1198772

2minus (119909119878minus 119909)2

119909119878= 119909119870

+119904119870

radic1 + 1199042

119870

1198772 119910

119878= 119910119870

minus1

radic1 + 1199042

119870

1198772

(50)

where 119909 gt 119909119870

In (49) and (50) 119909119870denotes the abscissae of the transition

curve end119910119870is the ordinate of the transition curve endwhile

119904119870is the value of tangent at the end of the transition curve

Values for numerical examples under consideration are givenin Table 1

Figures 7 and 8 illustrate the geometric systems made upof two circular arcs of consistent curvature connected withvarious types of transition curves As seen the application of agiven type of transition curve leads to an adequate position ofthe output circular arcThe position of this arc by the use of alinear transition curve clearly deviates from such cases whereadvantage is taken of curves with polynomial and trigono-metrical curvatureWith regard to the latter amutual positionof the circular arc varies insignificantly

Mathematical Problems in Engineering 11

Table 1 Comparison of values 119909119870 119910119870 and 119904

119870for numerical examples analyzed

Curve 119897119896[m] Θ(119897

119896) [rad] 119904

119870119909119870[m] 119910

119870[m]

Case 1 1198961= 11200 radm and 119896

2= 1700 radm

Curve C0 40 0045238 0045269 39988 0825Curve C1 (p) 60 0067857 0067961 59963 1821Curve C1 (t) 62832 0071060 0071180 62796 1940

Case 2 1198961= 11200 radm and 119896

2= minus1700 radm

Curve C0 60 minus0017857 minus0017859 59999 0143Curve C1 (p) 90 minus0026786 minus0026792 89991 0627Curve C1 (t) 94248 minus0028050 minus0028057 94238 0714

Case 3 1198961= minus11200 radm and 119896

2= 1700 radm

Curve C0 60 001786 0017859 59999 minus0143Curve C1 (p) 90 0026786 0026792 89991 minus0627Curve C1 (t) 94248 0028050 0028057 94238 minus0714

Case 4 1198961= minus11200 radm and 119896

2= minus1700 radm

Curve C0 40 minus0045238 minus0045269 39988 minus0825Curve C1 (p) 60 minus0067857 minus0067961 59969 minus1821Curve C1 (t) 62832 minus0071060 minus0071180 62976 minus1940

0

1

2

3

4

5

6

0 20 40 60 80 100 120x (m)

y(m

)

minus40 minus20

Figure 7 Examples of geometric systems consisting of two circulararcs with consistent curvature (brown line) connected with differenttransition curves for positive values of curvature (using differenthorizontal and vertical scales) curve of linear curvature in violetline curve of polynomial curvature in green and curve of trigono-metric curvature in red

00 20 40 60 80 100 120

x (m)

y(m

)

minus40 minus20

minus1

minus2

minus3

minus4

minus5

minus6

minus7

Figure 8 Examples of geometric systems consisting of two circulararcs with consistent curvature (brown line) connected with differenttransition curves for negative values of curvature (using differenthorizontal and vertical scales) curve of linear curvature in violetline curve of polynomial curvature in green and curve of trigono-metric curvature in red

0

05

1

15

0 50 100 150x (m)

y(m

)

minus05

minus1

minus15

minus2

minus50

Figure 9 Examples of geometric systems consisting of two inversecircular arcs (brown line) connected with different types of transi-tion curves for positive values of input curvature (using differenthorizontal and vertical scales) curve of linear curvature in violetline curve of polynomial curvature in green and curve of trigono-metric curvature in red

Figures 9 and 10 illustrate geometric systems made up oftwo inverse circular arcs connected with various types oftransition curves

Here the layout of the output circular arc is much morediversifiedThe position of the arc obtained by the applicationof the transition curve of linear curvature diverts by severaldozenmeters from situationswhere curves of polynomial andtrigonometric curvature have been used The difference forcurves of polynomial and trigonometric curvature attains inthe numerical example analyzed an order of several meters

The presented examples are taken in the very nature ofthings at random They reveal wide possibilities for the pre-sented analytical method in modelling geometric systems Itsapplication makes it possible to generate diversified layout ofthe communication routes adapted to the field conditions andother circumstances Simultaneously the method ensurescorrectness of the obtained solution in view of the kinematicparameters magnitudes in force

12 Mathematical Problems in Engineering

0

05

1

15

2

0 50 100 150x (m)

y(m

)

minus05

minus1

minus15

minus50

Figure 10 Examples of geometric systems consisting of twoinverse circular arcs (brown line) connected with different typesof transition curves for negative values of input curvature (usingdifferent horizontal and vertical scales) curve of linear curvaturein violet line curve of polynomial curvature in green and curve oftrigonometric curvature in red

7 Conclusions

The presented universal method of modelling geometricalsystem of a communication route is based on the determina-tion of an adequate curvature equation by using differentialequationsThe technique enables us to design connections ofvarious geometrical elements of the route indicating diversifi-cation in its curvature It is characterized by an analytical wayof recording certain functions and moreover it is possible tospecify in advance the class of these functions

The sought-after connection is recorded in the form ofparametric equations 119909(119897) and 119910(119897) where parameter 119897 isthe position of a given point along the length of the curveThe determination of 119909(119897) and 119910(119897) needs the expansion ofintegrand functions into Maclaurin series and with regard toinverse curvatures into Taylor series

The determination of coordinates of an adequate tran-sition curve in rectangular coordinate system 119909 119910 makesit possible for an easy transfer of it to Gauss-Kruger (119883 119884)

conformal coordinates and its subsequent layout in fieldmaking use of the satellite measuring technique GNSS

The method can be applied to identify various typesof transition curves The paper gives examples of solvingthe problem involving linear change of curvature and thecurvature in polynomial and trigonometric form

The performed comparative analysis reveals a wide rangeof possibilities which are offered by the presented designmethod Its application enables us to generate diversifiedlayout of the route It simultaneously ensures a completecontrol of correctness of the obtained solutions in view ofsatisfying the geometric and kinematic conditions

The presented method of modelling curvature has auniversal character It can be used for vehicular roads andrailway lines as well In the case of railway lines therearises an additional possibility of modelling in a similar waythe superelevation ramp understood here as a determineddifference in the height of the rail courses

Conflict of Interests

The author declares that he has no conflict of interestsregarding the publication of this paper

References

[1] W Koc and C Specht ldquoApplication of the Polish active GNSSgeodetic network for surveying and design of the railroadrdquo inProceedings of the 1st International Conference on Road and RailInfrastructure (CETRA rsquo10) pp 757ndash762 May 2010

[2] P Lipar M Lakner T Maher andM Zura ldquoEstimation of roadcenterline curvature from raw GPS datardquo Baltic Journal of Roadand Bridge Engineering vol 6 no 3 pp 163ndash168 2011

[3] L Tasci and N Kuloglu ldquoInvestigation of a new transitioncurverdquoTheBaltic Journal of Road and Bridge Engineering vol 6no 1 pp 23ndash29 2011

[4] H-H Cai and G-J Wang ldquoA new method in highway routedesign joining circular arcs by a single C-Bezier curve withshape parameterrdquo Journal of Zhejiang University Science A vol10 no 4 pp 562ndash569 2009

[5] S Dimulyo Z Habib and M Sakai ldquoFair cubic transition bet-ween two circles with one circle inside or tangent to the otherrdquoNumerical Algorithms vol 51 no 4 pp 461ndash476 2009

[6] G Farin ldquoClass A Bezier curvesrdquo Computer Aided GeometricDesign vol 23 no 7 pp 573ndash581 2006

[7] Z Habib and M Sakai ldquoG2 pythagorean hodograph quintictransition between two circles with shape controlrdquo ComputerAided Geometric Design vol 24 no 5 pp 252ndash266 2007

[8] G Harary and A Tal ldquoThe natural 3D spiralrdquo Computer Graph-ics Forum vol 30 no 2 pp 237ndash246 2011

[9] K T Miura ldquoA general equation of aesthetic curves and its self-affinityrdquo Computer-Aided Design and Applications vol 3 no 1ndash4 pp 457ndash464 2006

[10] S H Yahaya M S Salleh and J M Ali ldquoSpur gear design withan S-shaped transition curve application using mathematicaand CAD toolsrdquo in Proceedings of the International Conferenceon Computer Technology and Development (ICCTD rsquo09) pp273ndash276 Dubai United Arab Emirates November 2009

[11] N Yoshida and T Saito ldquoQuasi-aesthetic curves in rationalcubic Bezier formsrdquo Computer-Aided Design and Applicationsvol 4 no 1ndash6 pp 477ndash486 2007

[12] R Ziatdinov N Yoshida and T-W Kim ldquoAnalytic parametricequations of log-aesthetic curves in terms of incomplete gammafunctionsrdquo Computer Aided Geometric Design vol 29 no 2 pp129ndash140 2012

[13] R J Grabowski ldquoSmooth curvilinear transitions along vehicu-lar roads and railway linesrdquo Science Papers of AGH Universityof Science and Technology no 82 Cracow Poland 1984(Polish)

[14] L T Klauder ldquoTrack transition curve geometry based on gegen-bauer polynomialsrdquo in Proceedings of the 6th International Con-ference on Railway Engineering May 2003

[15] W Koc Elements of Design Theory on Track Systems GdanskUniversity of Technology Gdansk Poland 2004 (Polish)

[16] E Mieloszyk and W Koc ldquoGeneral dynamic method for det-ermining transition curve equationsrdquo Rail InternationalmdashSchienen der Welt vol 10 pp 32ndash40 1991

[17] B Kufver ldquoRealigning railway in track renewalsmdashlinear versusS-shaped superelevation rampsrdquo in Proceedings of the 2ndInternational Conference on Railway Engineering May 1999

[18] W Koc and E Mieloszyk ldquoComparative analysis of selectedtransition curves by the use of dynamicmodelrdquoArchives of CivilEngineering vol 33 no 2 pp 239ndash261 1987 (Polish)

[19] S Gleason and D Gebre-Egziabher GNSS Applications andMethods Artech House Publishers Norwood NJ USA 2009

Mathematical Problems in Engineering 13

[20] B Hoffmann-Wellenhof H Lichtenegger and E Wasle GNSSGlobal Navigation Satellite SystemsmdashGPS GLONASS Galileo ampMore Springer Wien Austria 2008

[21] P Misra and P Enge Global Positioning System Signals Mea-surements and Performance Ganga-Jamuna Press LincolnMass USA 2012

[22] C Specht ldquoAccuracy and coverage of the modernized PolishMaritime differential GPS systemrdquo Advances in Space Researchvol 47 no 2 pp 221ndash228 2011

[23] W Koc and K Palikowska ldquoDynamic properties evaluation ofthe selected methods of joining route segments with differentcurvaturerdquo Technika Transportu Szynowego vol 9 pp 1785ndash1807 2012 (Polish)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

119910 (119897) = int sinΘ (119897) 119889119897

=1

211989611198972minus [

1

241198963

1minus

1

48

1205872

1198972

119896

(1198962minus 1198961)] 1198974

+ [1

7201198965

1minus

1

1441198962

1

1205872

1198972

119896

(1198962minus 1198961)

minus1

1440

1205874

1198974

119896

(1198962minus 1198961)] 1198976

minus [1

403201198967

1minus

1

23041198964

1

1205872

1198972

119896

(1198962minus 1198961)

+1

2304

1205874

1198974

119896

1198961(1198962minus 1198961)2minus

1

3840

1205874

1198974

119896

1198962

1(1198962minus 1198961)

+1

80640

1205876

1198976

119896

(1198962minus 1198961)] 1198978+ sdot sdot sdot

(41)

Equations (41) for 1198961

= 0 describe the curve known ascosinusoid which can be used to connect a straight with acircular arc

53 Transition Curve Coordinates Connecting Inverse Curva-tures Near inverse arcs in the diagram of functionΘ(119897) thereappears the extremum at point 119897

0 where

Θ1015840(1198970) = 119896 (119897

0) =

1

2(1198961+ 1198962) minus

1

2(1198962minus 1198961) cos 120587

119897119896

1198970= 0

(42)

The transition curve parametric equations (41) are effec-tive for 119897 isin ⟨0 119897

0⟩ To determine value 119897

0one must solve (42)

The equation can be written in the form

cos 120587

119897119896

1198970=

1198961+ 1198962

1198962minus 1198961

(43)

It is easy to prove that for sign 1198961

= sign 1198962it is necessary

to satisfy the condition |1198961+1198962| lt |1198962minus1198961| thus |(119896

1+1198962)(1198962minus

1198961)| lt 1 The formula for length 119897

0is as follows

1198970=

1

120587(arccos1198961 + 119896

2

1198962minus 1198961

) 119897119896 (44)

For 119897 isin (1198970 119897119896⟩ functions cosΘ(119897) and sinΘ(119897) should

be expanded into Taylor series The integration provides thefollowing equations

119909 (119897) = int cosΘ (119897) 119889119897

= 119909 (1198970) + cosΘ(119897

0) (119897 minus 119897

0)

minus1

6sinΘ(119897

0)Θ10158401015840(1198970) (119897 minus 119897

0)3

minus1

24sinΘ(119897

0)Θ101584010158401015840

(1198970) sdot (119897 minus 119897

0)4

minus 1

40cosΘ(119897

0) [Θ10158401015840(1198970)]2

+1

120sinΘ(119897

0)Θ(4)

(1198970) (119897 minus 119897

0)5

minus 1

72cosΘ(119897

0)Θ10158401015840(1198970)Θ101584010158401015840

(1198970)

+1

720sinΘ(119897

0)Θ(5)

(1198970) (119897 minus 119897

0)6

+ 1

336sinΘ(119897

0) [Θ10158401015840(1198970)]3

minus1

336cosΘ(119897

0)Θ10158401015840(1198970)Θ(4)

(1198970)

minus1

504cosΘ(119897

0) [Θ101584010158401015840

(1198970)]2

minus1

5040sinΘ(119897

0)Θ(6)

(1198970) (119897 minus 119897

0)7

+ 1

384sinΘ(119897

0) [Θ10158401015840(1198970)]2

Θ101584010158401015840

(1198970)

minus1

1920cosΘ(119897

0)Θ10158401015840(119897) Θ(5)

(119897)

minus1

1152cosΘ(119897

0)Θ101584010158401015840

(1198970)Θ(4)

(1198970)

minus1

40320sinΘ(119897

0)Θ(7)

(1198970) (119897 minus 119897

0)8

+ 1

3456cosΘ(119897

0) [Θ10158401015840(1198970)]4

+1

1296sinΘ(119897

0)Θ10158401015840(1198970) [Θ101584010158401015840

(1198970)]2

+1

1728sinΘ(119897

0) [Θ10158401015840(1198970)]2

Θ(4)

(1198970)

minus1

12960cosΘ(119897

0)Θ10158401015840(1198970)Θ(6)

(1198970)

minus1

6480cosΘ(119897

0)Θ101584010158401015840

(1198970)Θ(5)

(1198970)

minus1

10368cosΘ(119897

0) [Θ(4)

(1198970)]2

minus1

362880sinΘ(119897

0)Θ(8)

(1198970) (119897 minus 119897

0)9+ sdot sdot sdot

Mathematical Problems in Engineering 9

119910 (119897) = int sinΘ (119897) 119889119897

= 119910 (1198970) + sinΘ(119897

0) (119897 minus 119897

0)

+1

6cosΘ(119897

0)Θ10158401015840(1198970) (119897 minus 119897

0)3

+1

24cosΘ(119897

0)Θ101584010158401015840

(1198970) sdot (119897 minus 119897

0)4

minus 1

40sinΘ(119897

0) [Θ10158401015840(1198970)]2

minus1

120cosΘ(119897

0)Θ(4)

(1198970) (119897 minus 119897

0)5

minus 1

72sinΘ(119897

0) 11987610158401015840(1198970)Θ101584010158401015840

(1198970)

minus1

720cosΘ(119897

0)Θ(5)

(1198970) (119897 minus 119897

0)6

minus 1

336cosΘ(119897

0) [Θ10158401015840(1198970)]3

+1

336sinΘ(119897

0)Θ10158401015840(1198970)Θ(4)

(1198970)

+1

504sinΘ(119897

0) [Θ101584010158401015840

(1198970)]2

minus1

5040cosΘ(119897

0)Θ(6)

(1198970) (119897 minus 119897

0)7

minus 1

384cosΘ(119897

0) [Θ10158401015840(1198970)]2

Θ101584010158401015840

(1198970)

+1

1920sinΘ(119897

0)Θ10158401015840(1198970)Θ(5)

(1198970)

+1

1152sinΘ(119897

0)Θ101584010158401015840

(1198970)Θ(4)

(1198970)

minus1

40320cosΘ(119897

0)Θ(7)

(1198970) (119897 minus 119897

0)8

+ 1

3456sinΘ(119897

0) [Θ10158401015840(1198970)]4

minus1

1296cosΘ(119897

0)Θ10158401015840(1198970) [Θ101584010158401015840

(1198970)]2

minus1

1728cosΘ(119897

0) [Θ10158401015840(1198970)]2

Θ(4)

(1198970)

minus1

12960sin119876 (119897

0) 11987610158401015840(1198970) 119876(6)

(1198970)

minus1

6480sin119876 (119897

0) 119876101584010158401015840

(1198970) 119876(5)

(1198970)

minus1

10368sinΘ(119897

0) [Θ(4)

(1198970)]2

+1

362880cosΘ(119897

0)Θ(8)

(1198970) (119897 minus 119897

0)9+ sdot sdot sdot

(45)

where

Θ(1198970) =

1

2(1198961+ 1198962) 1198970plusmn

119897119896

120587radicminus11989611198962

Θ10158401015840(1198970) = ∓

120587

119897119896

radicminus11989611198962

Θ10158401015840(1198970) = ∓

120587

119897119896

radicminus11989611198962

Θ101584010158401015840

(1198970) =

1

2

1205872

1198972

119896

(1198961+ 1198962)

Θ(4)

(1198970) = plusmn

1205873

1198973

119896

radicminus11989611198962

Θ(5)

(1198970) = minus

1

2

1205874

1198974

119896

(1198961+ 1198962)

Θ(6)

(1198970) = ∓

1205875

1198975

119896

radicminus11989611198962

Θ(7)

(1198970) =

1

2

1205876

1198976

119896

(1198961+ 1198962)

Θ(8)

(1198970) = plusmn

1205877

1198977

119896

radicminus11989611198962

(46)

In expressions with double signs plusmn and ∓ the upper onecorresponds to the case 119896

1gt 0 119896

2lt 0 while the lower one

refers to 1198961lt 0 119896

2gt 0

The value of tangent slope at the end of the transitioncurvemakes it possible to connect to it a curve of curvature 119896

2

satisfying the tangency condition of both the curves Thevalue is the same as in the previous cases It is determinedby (25)

6 Comparison of Horizontal Ordinates ofDetermined Transition Curves

61 The Shaping of Horizontal Ordinates of ComparableCurves If the analysis is carried out for railway routes thenthe necessity of meeting kinematic conditions (10) and (11)will require a diversification of the lengths of the comparedwith each other curves (using the same speed of trains)Moreover it is necessary to individually consider cases ofconnecting homogeneous curvatures (sign 119896

1= sign 119896

2) and

inverse arcs (sign 1198961

= sign 1198962)

It is assumed that at joints of consistent arcs with radii1198771= 1200m and 119877

2= 700m an attempt is made to attain a

speed V = 110 kmh Hence it is necessary to employ on arcssuperelevation of ℎ

1= 80mmand ℎ

2= 115mm respectively

The retention of the same kinematic parameters can providediversified lengths of the transition curve

(i) for a curve of linear curvature 119897119896= 40m

(ii) for a curve of polynomial curvature 119897119896= 60m

(iii) for a curve of trigonometric curvature 119897119896= 62 832m

10 Mathematical Problems in Engineering

The above length relations result directly from the equa-tions given at Sections 32 41 and 51

Figure 5 gives diagrams of horizontal ordinates for com-pared transition curves of positive and negative curvaturevalues As seen the curvature ordinates of linear curvatureevidently deviate from the ordinates of other curves In turncurves of polynomial and trigonometric curvature are inprinciple consistent A difference can only be noted in thefinal area And so the class of the function describing thecurvature plays here a significant part with respect to a curveof linear curvature it is a function of class C0 while regardingthe other two curves it is a function of class C1

At the joint of inverse arcs of radii 1198771= 1200m and 119877

2=

700m an attempt is made to reach the speed of V = 90 kmhHence it follows that it is necessary to apply superelevationalong arcs attaining ℎ

1= 20mm and ℎ

2= 45mm respec-

tively By the use the same kinematic parameters it is possibleto obtain the following lengths of the transition curve

(i) for a curve of linear curvature 119897119896= 60m

(ii) for a curve of polynomial curvature 119897119896= 90m

(iii) for a curve of trigonometric curvature 119897119896= 94 248m

Figure 6 presents diagrams of horizontal ordinates ofcompared transition curves for positive and negative valuesof curvature

As can be seen like in the case of consistent arcs the curveordinates of linear curvature deviate significantly from theordinates of the remaining curves but the shape of thesecurves also differs one from the other The class of functiondescribing the curvature is still important However in therange of the same class various geometric solutions arepossible

62 Negotiation of Transition Curves in a Geometric SystemThe transition curves do not appear alone in a geometricsystem but must be connected with some arcs in the neigh-borhood If the input arc is circular then its equation is asfollows

(i) for 1198961gt 0

119910 (119909) = 1198771minus radic1198772

1minus 1199092 (47)

(ii) for 1198961lt 0

119910 (119909) = minus (1198771minus radic1198772

1minus 1199092) (48)

where 119909 lt 0

However if the output arc is also a circular arc then it isexpressed in the following equation

(i) for 1198962gt 0

119910 (119909) = 119910119878minus radic1198772

2minus (119909119878minus 119909)2

119909119878= 119909119870

minus119904119870

radic1 + 1199042

119870

1198772 119910

119878= 119910119870

+1

radic1 + 1199042

119870

1198772

(49)

0051

152

25

0 10 20 30 40 50 60 70x (m)

y(m

)

minus05

minus1

minus15

minus2

minus25

Figure 5 Diagrams of horizontal ordinates of curves joiningconsistent arcs for positive and negative values of curvature (usingdifferent horizontal and vertical scales) curve of linear curvatureline in violet colour curve of polynomial curvature in green colourand curve of trigonometric curvature in red colour

0

05

1

15

0 20 40 60 80 100x (m)

y(m

)

minus05

minus1

minus15

Figure 6 Diagrams of horizontal ordinates of curves connectinginverse arcs for positive and negative values of curvature (usingdifferent horizontal and vertical scales) the curve of linear curvatureis in violet color the curve of polynomial curvature is in green colorand the curve of trigonometric curvature is in red color

(ii) for 1198962lt 0

119910 (119909) = 119910119878+ radic1198772

2minus (119909119878minus 119909)2

119909119878= 119909119870

+119904119870

radic1 + 1199042

119870

1198772 119910

119878= 119910119870

minus1

radic1 + 1199042

119870

1198772

(50)

where 119909 gt 119909119870

In (49) and (50) 119909119870denotes the abscissae of the transition

curve end119910119870is the ordinate of the transition curve endwhile

119904119870is the value of tangent at the end of the transition curve

Values for numerical examples under consideration are givenin Table 1

Figures 7 and 8 illustrate the geometric systems made upof two circular arcs of consistent curvature connected withvarious types of transition curves As seen the application of agiven type of transition curve leads to an adequate position ofthe output circular arcThe position of this arc by the use of alinear transition curve clearly deviates from such cases whereadvantage is taken of curves with polynomial and trigono-metrical curvatureWith regard to the latter amutual positionof the circular arc varies insignificantly

Mathematical Problems in Engineering 11

Table 1 Comparison of values 119909119870 119910119870 and 119904

119870for numerical examples analyzed

Curve 119897119896[m] Θ(119897

119896) [rad] 119904

119870119909119870[m] 119910

119870[m]

Case 1 1198961= 11200 radm and 119896

2= 1700 radm

Curve C0 40 0045238 0045269 39988 0825Curve C1 (p) 60 0067857 0067961 59963 1821Curve C1 (t) 62832 0071060 0071180 62796 1940

Case 2 1198961= 11200 radm and 119896

2= minus1700 radm

Curve C0 60 minus0017857 minus0017859 59999 0143Curve C1 (p) 90 minus0026786 minus0026792 89991 0627Curve C1 (t) 94248 minus0028050 minus0028057 94238 0714

Case 3 1198961= minus11200 radm and 119896

2= 1700 radm

Curve C0 60 001786 0017859 59999 minus0143Curve C1 (p) 90 0026786 0026792 89991 minus0627Curve C1 (t) 94248 0028050 0028057 94238 minus0714

Case 4 1198961= minus11200 radm and 119896

2= minus1700 radm

Curve C0 40 minus0045238 minus0045269 39988 minus0825Curve C1 (p) 60 minus0067857 minus0067961 59969 minus1821Curve C1 (t) 62832 minus0071060 minus0071180 62976 minus1940

0

1

2

3

4

5

6

0 20 40 60 80 100 120x (m)

y(m

)

minus40 minus20

Figure 7 Examples of geometric systems consisting of two circulararcs with consistent curvature (brown line) connected with differenttransition curves for positive values of curvature (using differenthorizontal and vertical scales) curve of linear curvature in violetline curve of polynomial curvature in green and curve of trigono-metric curvature in red

00 20 40 60 80 100 120

x (m)

y(m

)

minus40 minus20

minus1

minus2

minus3

minus4

minus5

minus6

minus7

Figure 8 Examples of geometric systems consisting of two circulararcs with consistent curvature (brown line) connected with differenttransition curves for negative values of curvature (using differenthorizontal and vertical scales) curve of linear curvature in violetline curve of polynomial curvature in green and curve of trigono-metric curvature in red

0

05

1

15

0 50 100 150x (m)

y(m

)

minus05

minus1

minus15

minus2

minus50

Figure 9 Examples of geometric systems consisting of two inversecircular arcs (brown line) connected with different types of transi-tion curves for positive values of input curvature (using differenthorizontal and vertical scales) curve of linear curvature in violetline curve of polynomial curvature in green and curve of trigono-metric curvature in red

Figures 9 and 10 illustrate geometric systems made up oftwo inverse circular arcs connected with various types oftransition curves

Here the layout of the output circular arc is much morediversifiedThe position of the arc obtained by the applicationof the transition curve of linear curvature diverts by severaldozenmeters from situationswhere curves of polynomial andtrigonometric curvature have been used The difference forcurves of polynomial and trigonometric curvature attains inthe numerical example analyzed an order of several meters

The presented examples are taken in the very nature ofthings at random They reveal wide possibilities for the pre-sented analytical method in modelling geometric systems Itsapplication makes it possible to generate diversified layout ofthe communication routes adapted to the field conditions andother circumstances Simultaneously the method ensurescorrectness of the obtained solution in view of the kinematicparameters magnitudes in force

12 Mathematical Problems in Engineering

0

05

1

15

2

0 50 100 150x (m)

y(m

)

minus05

minus1

minus15

minus50

Figure 10 Examples of geometric systems consisting of twoinverse circular arcs (brown line) connected with different typesof transition curves for negative values of input curvature (usingdifferent horizontal and vertical scales) curve of linear curvaturein violet line curve of polynomial curvature in green and curve oftrigonometric curvature in red

7 Conclusions

The presented universal method of modelling geometricalsystem of a communication route is based on the determina-tion of an adequate curvature equation by using differentialequationsThe technique enables us to design connections ofvarious geometrical elements of the route indicating diversifi-cation in its curvature It is characterized by an analytical wayof recording certain functions and moreover it is possible tospecify in advance the class of these functions

The sought-after connection is recorded in the form ofparametric equations 119909(119897) and 119910(119897) where parameter 119897 isthe position of a given point along the length of the curveThe determination of 119909(119897) and 119910(119897) needs the expansion ofintegrand functions into Maclaurin series and with regard toinverse curvatures into Taylor series

The determination of coordinates of an adequate tran-sition curve in rectangular coordinate system 119909 119910 makesit possible for an easy transfer of it to Gauss-Kruger (119883 119884)

conformal coordinates and its subsequent layout in fieldmaking use of the satellite measuring technique GNSS

The method can be applied to identify various typesof transition curves The paper gives examples of solvingthe problem involving linear change of curvature and thecurvature in polynomial and trigonometric form

The performed comparative analysis reveals a wide rangeof possibilities which are offered by the presented designmethod Its application enables us to generate diversifiedlayout of the route It simultaneously ensures a completecontrol of correctness of the obtained solutions in view ofsatisfying the geometric and kinematic conditions

The presented method of modelling curvature has auniversal character It can be used for vehicular roads andrailway lines as well In the case of railway lines therearises an additional possibility of modelling in a similar waythe superelevation ramp understood here as a determineddifference in the height of the rail courses

Conflict of Interests

The author declares that he has no conflict of interestsregarding the publication of this paper

References

[1] W Koc and C Specht ldquoApplication of the Polish active GNSSgeodetic network for surveying and design of the railroadrdquo inProceedings of the 1st International Conference on Road and RailInfrastructure (CETRA rsquo10) pp 757ndash762 May 2010

[2] P Lipar M Lakner T Maher andM Zura ldquoEstimation of roadcenterline curvature from raw GPS datardquo Baltic Journal of Roadand Bridge Engineering vol 6 no 3 pp 163ndash168 2011

[3] L Tasci and N Kuloglu ldquoInvestigation of a new transitioncurverdquoTheBaltic Journal of Road and Bridge Engineering vol 6no 1 pp 23ndash29 2011

[4] H-H Cai and G-J Wang ldquoA new method in highway routedesign joining circular arcs by a single C-Bezier curve withshape parameterrdquo Journal of Zhejiang University Science A vol10 no 4 pp 562ndash569 2009

[5] S Dimulyo Z Habib and M Sakai ldquoFair cubic transition bet-ween two circles with one circle inside or tangent to the otherrdquoNumerical Algorithms vol 51 no 4 pp 461ndash476 2009

[6] G Farin ldquoClass A Bezier curvesrdquo Computer Aided GeometricDesign vol 23 no 7 pp 573ndash581 2006

[7] Z Habib and M Sakai ldquoG2 pythagorean hodograph quintictransition between two circles with shape controlrdquo ComputerAided Geometric Design vol 24 no 5 pp 252ndash266 2007

[8] G Harary and A Tal ldquoThe natural 3D spiralrdquo Computer Graph-ics Forum vol 30 no 2 pp 237ndash246 2011

[9] K T Miura ldquoA general equation of aesthetic curves and its self-affinityrdquo Computer-Aided Design and Applications vol 3 no 1ndash4 pp 457ndash464 2006

[10] S H Yahaya M S Salleh and J M Ali ldquoSpur gear design withan S-shaped transition curve application using mathematicaand CAD toolsrdquo in Proceedings of the International Conferenceon Computer Technology and Development (ICCTD rsquo09) pp273ndash276 Dubai United Arab Emirates November 2009

[11] N Yoshida and T Saito ldquoQuasi-aesthetic curves in rationalcubic Bezier formsrdquo Computer-Aided Design and Applicationsvol 4 no 1ndash6 pp 477ndash486 2007

[12] R Ziatdinov N Yoshida and T-W Kim ldquoAnalytic parametricequations of log-aesthetic curves in terms of incomplete gammafunctionsrdquo Computer Aided Geometric Design vol 29 no 2 pp129ndash140 2012

[13] R J Grabowski ldquoSmooth curvilinear transitions along vehicu-lar roads and railway linesrdquo Science Papers of AGH Universityof Science and Technology no 82 Cracow Poland 1984(Polish)

[14] L T Klauder ldquoTrack transition curve geometry based on gegen-bauer polynomialsrdquo in Proceedings of the 6th International Con-ference on Railway Engineering May 2003

[15] W Koc Elements of Design Theory on Track Systems GdanskUniversity of Technology Gdansk Poland 2004 (Polish)

[16] E Mieloszyk and W Koc ldquoGeneral dynamic method for det-ermining transition curve equationsrdquo Rail InternationalmdashSchienen der Welt vol 10 pp 32ndash40 1991

[17] B Kufver ldquoRealigning railway in track renewalsmdashlinear versusS-shaped superelevation rampsrdquo in Proceedings of the 2ndInternational Conference on Railway Engineering May 1999

[18] W Koc and E Mieloszyk ldquoComparative analysis of selectedtransition curves by the use of dynamicmodelrdquoArchives of CivilEngineering vol 33 no 2 pp 239ndash261 1987 (Polish)

[19] S Gleason and D Gebre-Egziabher GNSS Applications andMethods Artech House Publishers Norwood NJ USA 2009

Mathematical Problems in Engineering 13

[20] B Hoffmann-Wellenhof H Lichtenegger and E Wasle GNSSGlobal Navigation Satellite SystemsmdashGPS GLONASS Galileo ampMore Springer Wien Austria 2008

[21] P Misra and P Enge Global Positioning System Signals Mea-surements and Performance Ganga-Jamuna Press LincolnMass USA 2012

[22] C Specht ldquoAccuracy and coverage of the modernized PolishMaritime differential GPS systemrdquo Advances in Space Researchvol 47 no 2 pp 221ndash228 2011

[23] W Koc and K Palikowska ldquoDynamic properties evaluation ofthe selected methods of joining route segments with differentcurvaturerdquo Technika Transportu Szynowego vol 9 pp 1785ndash1807 2012 (Polish)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

119910 (119897) = int sinΘ (119897) 119889119897

= 119910 (1198970) + sinΘ(119897

0) (119897 minus 119897

0)

+1

6cosΘ(119897

0)Θ10158401015840(1198970) (119897 minus 119897

0)3

+1

24cosΘ(119897

0)Θ101584010158401015840

(1198970) sdot (119897 minus 119897

0)4

minus 1

40sinΘ(119897

0) [Θ10158401015840(1198970)]2

minus1

120cosΘ(119897

0)Θ(4)

(1198970) (119897 minus 119897

0)5

minus 1

72sinΘ(119897

0) 11987610158401015840(1198970)Θ101584010158401015840

(1198970)

minus1

720cosΘ(119897

0)Θ(5)

(1198970) (119897 minus 119897

0)6

minus 1

336cosΘ(119897

0) [Θ10158401015840(1198970)]3

+1

336sinΘ(119897

0)Θ10158401015840(1198970)Θ(4)

(1198970)

+1

504sinΘ(119897

0) [Θ101584010158401015840

(1198970)]2

minus1

5040cosΘ(119897

0)Θ(6)

(1198970) (119897 minus 119897

0)7

minus 1

384cosΘ(119897

0) [Θ10158401015840(1198970)]2

Θ101584010158401015840

(1198970)

+1

1920sinΘ(119897

0)Θ10158401015840(1198970)Θ(5)

(1198970)

+1

1152sinΘ(119897

0)Θ101584010158401015840

(1198970)Θ(4)

(1198970)

minus1

40320cosΘ(119897

0)Θ(7)

(1198970) (119897 minus 119897

0)8

+ 1

3456sinΘ(119897

0) [Θ10158401015840(1198970)]4

minus1

1296cosΘ(119897

0)Θ10158401015840(1198970) [Θ101584010158401015840

(1198970)]2

minus1

1728cosΘ(119897

0) [Θ10158401015840(1198970)]2

Θ(4)

(1198970)

minus1

12960sin119876 (119897

0) 11987610158401015840(1198970) 119876(6)

(1198970)

minus1

6480sin119876 (119897

0) 119876101584010158401015840

(1198970) 119876(5)

(1198970)

minus1

10368sinΘ(119897

0) [Θ(4)

(1198970)]2

+1

362880cosΘ(119897

0)Θ(8)

(1198970) (119897 minus 119897

0)9+ sdot sdot sdot

(45)

where

Θ(1198970) =

1

2(1198961+ 1198962) 1198970plusmn

119897119896

120587radicminus11989611198962

Θ10158401015840(1198970) = ∓

120587

119897119896

radicminus11989611198962

Θ10158401015840(1198970) = ∓

120587

119897119896

radicminus11989611198962

Θ101584010158401015840

(1198970) =

1

2

1205872

1198972

119896

(1198961+ 1198962)

Θ(4)

(1198970) = plusmn

1205873

1198973

119896

radicminus11989611198962

Θ(5)

(1198970) = minus

1

2

1205874

1198974

119896

(1198961+ 1198962)

Θ(6)

(1198970) = ∓

1205875

1198975

119896

radicminus11989611198962

Θ(7)

(1198970) =

1

2

1205876

1198976

119896

(1198961+ 1198962)

Θ(8)

(1198970) = plusmn

1205877

1198977

119896

radicminus11989611198962

(46)

In expressions with double signs plusmn and ∓ the upper onecorresponds to the case 119896

1gt 0 119896

2lt 0 while the lower one

refers to 1198961lt 0 119896

2gt 0

The value of tangent slope at the end of the transitioncurvemakes it possible to connect to it a curve of curvature 119896

2

satisfying the tangency condition of both the curves Thevalue is the same as in the previous cases It is determinedby (25)

6 Comparison of Horizontal Ordinates ofDetermined Transition Curves

61 The Shaping of Horizontal Ordinates of ComparableCurves If the analysis is carried out for railway routes thenthe necessity of meeting kinematic conditions (10) and (11)will require a diversification of the lengths of the comparedwith each other curves (using the same speed of trains)Moreover it is necessary to individually consider cases ofconnecting homogeneous curvatures (sign 119896

1= sign 119896

2) and

inverse arcs (sign 1198961

= sign 1198962)

It is assumed that at joints of consistent arcs with radii1198771= 1200m and 119877

2= 700m an attempt is made to attain a

speed V = 110 kmh Hence it is necessary to employ on arcssuperelevation of ℎ

1= 80mmand ℎ

2= 115mm respectively

The retention of the same kinematic parameters can providediversified lengths of the transition curve

(i) for a curve of linear curvature 119897119896= 40m

(ii) for a curve of polynomial curvature 119897119896= 60m

(iii) for a curve of trigonometric curvature 119897119896= 62 832m

10 Mathematical Problems in Engineering

The above length relations result directly from the equa-tions given at Sections 32 41 and 51

Figure 5 gives diagrams of horizontal ordinates for com-pared transition curves of positive and negative curvaturevalues As seen the curvature ordinates of linear curvatureevidently deviate from the ordinates of other curves In turncurves of polynomial and trigonometric curvature are inprinciple consistent A difference can only be noted in thefinal area And so the class of the function describing thecurvature plays here a significant part with respect to a curveof linear curvature it is a function of class C0 while regardingthe other two curves it is a function of class C1

At the joint of inverse arcs of radii 1198771= 1200m and 119877

2=

700m an attempt is made to reach the speed of V = 90 kmhHence it follows that it is necessary to apply superelevationalong arcs attaining ℎ

1= 20mm and ℎ

2= 45mm respec-

tively By the use the same kinematic parameters it is possibleto obtain the following lengths of the transition curve

(i) for a curve of linear curvature 119897119896= 60m

(ii) for a curve of polynomial curvature 119897119896= 90m

(iii) for a curve of trigonometric curvature 119897119896= 94 248m

Figure 6 presents diagrams of horizontal ordinates ofcompared transition curves for positive and negative valuesof curvature

As can be seen like in the case of consistent arcs the curveordinates of linear curvature deviate significantly from theordinates of the remaining curves but the shape of thesecurves also differs one from the other The class of functiondescribing the curvature is still important However in therange of the same class various geometric solutions arepossible

62 Negotiation of Transition Curves in a Geometric SystemThe transition curves do not appear alone in a geometricsystem but must be connected with some arcs in the neigh-borhood If the input arc is circular then its equation is asfollows

(i) for 1198961gt 0

119910 (119909) = 1198771minus radic1198772

1minus 1199092 (47)

(ii) for 1198961lt 0

119910 (119909) = minus (1198771minus radic1198772

1minus 1199092) (48)

where 119909 lt 0

However if the output arc is also a circular arc then it isexpressed in the following equation

(i) for 1198962gt 0

119910 (119909) = 119910119878minus radic1198772

2minus (119909119878minus 119909)2

119909119878= 119909119870

minus119904119870

radic1 + 1199042

119870

1198772 119910

119878= 119910119870

+1

radic1 + 1199042

119870

1198772

(49)

0051

152

25

0 10 20 30 40 50 60 70x (m)

y(m

)

minus05

minus1

minus15

minus2

minus25

Figure 5 Diagrams of horizontal ordinates of curves joiningconsistent arcs for positive and negative values of curvature (usingdifferent horizontal and vertical scales) curve of linear curvatureline in violet colour curve of polynomial curvature in green colourand curve of trigonometric curvature in red colour

0

05

1

15

0 20 40 60 80 100x (m)

y(m

)

minus05

minus1

minus15

Figure 6 Diagrams of horizontal ordinates of curves connectinginverse arcs for positive and negative values of curvature (usingdifferent horizontal and vertical scales) the curve of linear curvatureis in violet color the curve of polynomial curvature is in green colorand the curve of trigonometric curvature is in red color

(ii) for 1198962lt 0

119910 (119909) = 119910119878+ radic1198772

2minus (119909119878minus 119909)2

119909119878= 119909119870

+119904119870

radic1 + 1199042

119870

1198772 119910

119878= 119910119870

minus1

radic1 + 1199042

119870

1198772

(50)

where 119909 gt 119909119870

In (49) and (50) 119909119870denotes the abscissae of the transition

curve end119910119870is the ordinate of the transition curve endwhile

119904119870is the value of tangent at the end of the transition curve

Values for numerical examples under consideration are givenin Table 1

Figures 7 and 8 illustrate the geometric systems made upof two circular arcs of consistent curvature connected withvarious types of transition curves As seen the application of agiven type of transition curve leads to an adequate position ofthe output circular arcThe position of this arc by the use of alinear transition curve clearly deviates from such cases whereadvantage is taken of curves with polynomial and trigono-metrical curvatureWith regard to the latter amutual positionof the circular arc varies insignificantly

Mathematical Problems in Engineering 11

Table 1 Comparison of values 119909119870 119910119870 and 119904

119870for numerical examples analyzed

Curve 119897119896[m] Θ(119897

119896) [rad] 119904

119870119909119870[m] 119910

119870[m]

Case 1 1198961= 11200 radm and 119896

2= 1700 radm

Curve C0 40 0045238 0045269 39988 0825Curve C1 (p) 60 0067857 0067961 59963 1821Curve C1 (t) 62832 0071060 0071180 62796 1940

Case 2 1198961= 11200 radm and 119896

2= minus1700 radm

Curve C0 60 minus0017857 minus0017859 59999 0143Curve C1 (p) 90 minus0026786 minus0026792 89991 0627Curve C1 (t) 94248 minus0028050 minus0028057 94238 0714

Case 3 1198961= minus11200 radm and 119896

2= 1700 radm

Curve C0 60 001786 0017859 59999 minus0143Curve C1 (p) 90 0026786 0026792 89991 minus0627Curve C1 (t) 94248 0028050 0028057 94238 minus0714

Case 4 1198961= minus11200 radm and 119896

2= minus1700 radm

Curve C0 40 minus0045238 minus0045269 39988 minus0825Curve C1 (p) 60 minus0067857 minus0067961 59969 minus1821Curve C1 (t) 62832 minus0071060 minus0071180 62976 minus1940

0

1

2

3

4

5

6

0 20 40 60 80 100 120x (m)

y(m

)

minus40 minus20

Figure 7 Examples of geometric systems consisting of two circulararcs with consistent curvature (brown line) connected with differenttransition curves for positive values of curvature (using differenthorizontal and vertical scales) curve of linear curvature in violetline curve of polynomial curvature in green and curve of trigono-metric curvature in red

00 20 40 60 80 100 120

x (m)

y(m

)

minus40 minus20

minus1

minus2

minus3

minus4

minus5

minus6

minus7

Figure 8 Examples of geometric systems consisting of two circulararcs with consistent curvature (brown line) connected with differenttransition curves for negative values of curvature (using differenthorizontal and vertical scales) curve of linear curvature in violetline curve of polynomial curvature in green and curve of trigono-metric curvature in red

0

05

1

15

0 50 100 150x (m)

y(m

)

minus05

minus1

minus15

minus2

minus50

Figure 9 Examples of geometric systems consisting of two inversecircular arcs (brown line) connected with different types of transi-tion curves for positive values of input curvature (using differenthorizontal and vertical scales) curve of linear curvature in violetline curve of polynomial curvature in green and curve of trigono-metric curvature in red

Figures 9 and 10 illustrate geometric systems made up oftwo inverse circular arcs connected with various types oftransition curves

Here the layout of the output circular arc is much morediversifiedThe position of the arc obtained by the applicationof the transition curve of linear curvature diverts by severaldozenmeters from situationswhere curves of polynomial andtrigonometric curvature have been used The difference forcurves of polynomial and trigonometric curvature attains inthe numerical example analyzed an order of several meters

The presented examples are taken in the very nature ofthings at random They reveal wide possibilities for the pre-sented analytical method in modelling geometric systems Itsapplication makes it possible to generate diversified layout ofthe communication routes adapted to the field conditions andother circumstances Simultaneously the method ensurescorrectness of the obtained solution in view of the kinematicparameters magnitudes in force

12 Mathematical Problems in Engineering

0

05

1

15

2

0 50 100 150x (m)

y(m

)

minus05

minus1

minus15

minus50

Figure 10 Examples of geometric systems consisting of twoinverse circular arcs (brown line) connected with different typesof transition curves for negative values of input curvature (usingdifferent horizontal and vertical scales) curve of linear curvaturein violet line curve of polynomial curvature in green and curve oftrigonometric curvature in red

7 Conclusions

The presented universal method of modelling geometricalsystem of a communication route is based on the determina-tion of an adequate curvature equation by using differentialequationsThe technique enables us to design connections ofvarious geometrical elements of the route indicating diversifi-cation in its curvature It is characterized by an analytical wayof recording certain functions and moreover it is possible tospecify in advance the class of these functions

The sought-after connection is recorded in the form ofparametric equations 119909(119897) and 119910(119897) where parameter 119897 isthe position of a given point along the length of the curveThe determination of 119909(119897) and 119910(119897) needs the expansion ofintegrand functions into Maclaurin series and with regard toinverse curvatures into Taylor series

The determination of coordinates of an adequate tran-sition curve in rectangular coordinate system 119909 119910 makesit possible for an easy transfer of it to Gauss-Kruger (119883 119884)

conformal coordinates and its subsequent layout in fieldmaking use of the satellite measuring technique GNSS

The method can be applied to identify various typesof transition curves The paper gives examples of solvingthe problem involving linear change of curvature and thecurvature in polynomial and trigonometric form

The performed comparative analysis reveals a wide rangeof possibilities which are offered by the presented designmethod Its application enables us to generate diversifiedlayout of the route It simultaneously ensures a completecontrol of correctness of the obtained solutions in view ofsatisfying the geometric and kinematic conditions

The presented method of modelling curvature has auniversal character It can be used for vehicular roads andrailway lines as well In the case of railway lines therearises an additional possibility of modelling in a similar waythe superelevation ramp understood here as a determineddifference in the height of the rail courses

Conflict of Interests

The author declares that he has no conflict of interestsregarding the publication of this paper

References

[1] W Koc and C Specht ldquoApplication of the Polish active GNSSgeodetic network for surveying and design of the railroadrdquo inProceedings of the 1st International Conference on Road and RailInfrastructure (CETRA rsquo10) pp 757ndash762 May 2010

[2] P Lipar M Lakner T Maher andM Zura ldquoEstimation of roadcenterline curvature from raw GPS datardquo Baltic Journal of Roadand Bridge Engineering vol 6 no 3 pp 163ndash168 2011

[3] L Tasci and N Kuloglu ldquoInvestigation of a new transitioncurverdquoTheBaltic Journal of Road and Bridge Engineering vol 6no 1 pp 23ndash29 2011

[4] H-H Cai and G-J Wang ldquoA new method in highway routedesign joining circular arcs by a single C-Bezier curve withshape parameterrdquo Journal of Zhejiang University Science A vol10 no 4 pp 562ndash569 2009

[5] S Dimulyo Z Habib and M Sakai ldquoFair cubic transition bet-ween two circles with one circle inside or tangent to the otherrdquoNumerical Algorithms vol 51 no 4 pp 461ndash476 2009

[6] G Farin ldquoClass A Bezier curvesrdquo Computer Aided GeometricDesign vol 23 no 7 pp 573ndash581 2006

[7] Z Habib and M Sakai ldquoG2 pythagorean hodograph quintictransition between two circles with shape controlrdquo ComputerAided Geometric Design vol 24 no 5 pp 252ndash266 2007

[8] G Harary and A Tal ldquoThe natural 3D spiralrdquo Computer Graph-ics Forum vol 30 no 2 pp 237ndash246 2011

[9] K T Miura ldquoA general equation of aesthetic curves and its self-affinityrdquo Computer-Aided Design and Applications vol 3 no 1ndash4 pp 457ndash464 2006

[10] S H Yahaya M S Salleh and J M Ali ldquoSpur gear design withan S-shaped transition curve application using mathematicaand CAD toolsrdquo in Proceedings of the International Conferenceon Computer Technology and Development (ICCTD rsquo09) pp273ndash276 Dubai United Arab Emirates November 2009

[11] N Yoshida and T Saito ldquoQuasi-aesthetic curves in rationalcubic Bezier formsrdquo Computer-Aided Design and Applicationsvol 4 no 1ndash6 pp 477ndash486 2007

[12] R Ziatdinov N Yoshida and T-W Kim ldquoAnalytic parametricequations of log-aesthetic curves in terms of incomplete gammafunctionsrdquo Computer Aided Geometric Design vol 29 no 2 pp129ndash140 2012

[13] R J Grabowski ldquoSmooth curvilinear transitions along vehicu-lar roads and railway linesrdquo Science Papers of AGH Universityof Science and Technology no 82 Cracow Poland 1984(Polish)

[14] L T Klauder ldquoTrack transition curve geometry based on gegen-bauer polynomialsrdquo in Proceedings of the 6th International Con-ference on Railway Engineering May 2003

[15] W Koc Elements of Design Theory on Track Systems GdanskUniversity of Technology Gdansk Poland 2004 (Polish)

[16] E Mieloszyk and W Koc ldquoGeneral dynamic method for det-ermining transition curve equationsrdquo Rail InternationalmdashSchienen der Welt vol 10 pp 32ndash40 1991

[17] B Kufver ldquoRealigning railway in track renewalsmdashlinear versusS-shaped superelevation rampsrdquo in Proceedings of the 2ndInternational Conference on Railway Engineering May 1999

[18] W Koc and E Mieloszyk ldquoComparative analysis of selectedtransition curves by the use of dynamicmodelrdquoArchives of CivilEngineering vol 33 no 2 pp 239ndash261 1987 (Polish)

[19] S Gleason and D Gebre-Egziabher GNSS Applications andMethods Artech House Publishers Norwood NJ USA 2009

Mathematical Problems in Engineering 13

[20] B Hoffmann-Wellenhof H Lichtenegger and E Wasle GNSSGlobal Navigation Satellite SystemsmdashGPS GLONASS Galileo ampMore Springer Wien Austria 2008

[21] P Misra and P Enge Global Positioning System Signals Mea-surements and Performance Ganga-Jamuna Press LincolnMass USA 2012

[22] C Specht ldquoAccuracy and coverage of the modernized PolishMaritime differential GPS systemrdquo Advances in Space Researchvol 47 no 2 pp 221ndash228 2011

[23] W Koc and K Palikowska ldquoDynamic properties evaluation ofthe selected methods of joining route segments with differentcurvaturerdquo Technika Transportu Szynowego vol 9 pp 1785ndash1807 2012 (Polish)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

The above length relations result directly from the equa-tions given at Sections 32 41 and 51

Figure 5 gives diagrams of horizontal ordinates for com-pared transition curves of positive and negative curvaturevalues As seen the curvature ordinates of linear curvatureevidently deviate from the ordinates of other curves In turncurves of polynomial and trigonometric curvature are inprinciple consistent A difference can only be noted in thefinal area And so the class of the function describing thecurvature plays here a significant part with respect to a curveof linear curvature it is a function of class C0 while regardingthe other two curves it is a function of class C1

At the joint of inverse arcs of radii 1198771= 1200m and 119877

2=

700m an attempt is made to reach the speed of V = 90 kmhHence it follows that it is necessary to apply superelevationalong arcs attaining ℎ

1= 20mm and ℎ

2= 45mm respec-

tively By the use the same kinematic parameters it is possibleto obtain the following lengths of the transition curve

(i) for a curve of linear curvature 119897119896= 60m

(ii) for a curve of polynomial curvature 119897119896= 90m

(iii) for a curve of trigonometric curvature 119897119896= 94 248m

Figure 6 presents diagrams of horizontal ordinates ofcompared transition curves for positive and negative valuesof curvature

As can be seen like in the case of consistent arcs the curveordinates of linear curvature deviate significantly from theordinates of the remaining curves but the shape of thesecurves also differs one from the other The class of functiondescribing the curvature is still important However in therange of the same class various geometric solutions arepossible

62 Negotiation of Transition Curves in a Geometric SystemThe transition curves do not appear alone in a geometricsystem but must be connected with some arcs in the neigh-borhood If the input arc is circular then its equation is asfollows

(i) for 1198961gt 0

119910 (119909) = 1198771minus radic1198772

1minus 1199092 (47)

(ii) for 1198961lt 0

119910 (119909) = minus (1198771minus radic1198772

1minus 1199092) (48)

where 119909 lt 0

However if the output arc is also a circular arc then it isexpressed in the following equation

(i) for 1198962gt 0

119910 (119909) = 119910119878minus radic1198772

2minus (119909119878minus 119909)2

119909119878= 119909119870

minus119904119870

radic1 + 1199042

119870

1198772 119910

119878= 119910119870

+1

radic1 + 1199042

119870

1198772

(49)

0051

152

25

0 10 20 30 40 50 60 70x (m)

y(m

)

minus05

minus1

minus15

minus2

minus25

Figure 5 Diagrams of horizontal ordinates of curves joiningconsistent arcs for positive and negative values of curvature (usingdifferent horizontal and vertical scales) curve of linear curvatureline in violet colour curve of polynomial curvature in green colourand curve of trigonometric curvature in red colour

0

05

1

15

0 20 40 60 80 100x (m)

y(m

)

minus05

minus1

minus15

Figure 6 Diagrams of horizontal ordinates of curves connectinginverse arcs for positive and negative values of curvature (usingdifferent horizontal and vertical scales) the curve of linear curvatureis in violet color the curve of polynomial curvature is in green colorand the curve of trigonometric curvature is in red color

(ii) for 1198962lt 0

119910 (119909) = 119910119878+ radic1198772

2minus (119909119878minus 119909)2

119909119878= 119909119870

+119904119870

radic1 + 1199042

119870

1198772 119910

119878= 119910119870

minus1

radic1 + 1199042

119870

1198772

(50)

where 119909 gt 119909119870

In (49) and (50) 119909119870denotes the abscissae of the transition

curve end119910119870is the ordinate of the transition curve endwhile

119904119870is the value of tangent at the end of the transition curve

Values for numerical examples under consideration are givenin Table 1

Figures 7 and 8 illustrate the geometric systems made upof two circular arcs of consistent curvature connected withvarious types of transition curves As seen the application of agiven type of transition curve leads to an adequate position ofthe output circular arcThe position of this arc by the use of alinear transition curve clearly deviates from such cases whereadvantage is taken of curves with polynomial and trigono-metrical curvatureWith regard to the latter amutual positionof the circular arc varies insignificantly

Mathematical Problems in Engineering 11

Table 1 Comparison of values 119909119870 119910119870 and 119904

119870for numerical examples analyzed

Curve 119897119896[m] Θ(119897

119896) [rad] 119904

119870119909119870[m] 119910

119870[m]

Case 1 1198961= 11200 radm and 119896

2= 1700 radm

Curve C0 40 0045238 0045269 39988 0825Curve C1 (p) 60 0067857 0067961 59963 1821Curve C1 (t) 62832 0071060 0071180 62796 1940

Case 2 1198961= 11200 radm and 119896

2= minus1700 radm

Curve C0 60 minus0017857 minus0017859 59999 0143Curve C1 (p) 90 minus0026786 minus0026792 89991 0627Curve C1 (t) 94248 minus0028050 minus0028057 94238 0714

Case 3 1198961= minus11200 radm and 119896

2= 1700 radm

Curve C0 60 001786 0017859 59999 minus0143Curve C1 (p) 90 0026786 0026792 89991 minus0627Curve C1 (t) 94248 0028050 0028057 94238 minus0714

Case 4 1198961= minus11200 radm and 119896

2= minus1700 radm

Curve C0 40 minus0045238 minus0045269 39988 minus0825Curve C1 (p) 60 minus0067857 minus0067961 59969 minus1821Curve C1 (t) 62832 minus0071060 minus0071180 62976 minus1940

0

1

2

3

4

5

6

0 20 40 60 80 100 120x (m)

y(m

)

minus40 minus20

Figure 7 Examples of geometric systems consisting of two circulararcs with consistent curvature (brown line) connected with differenttransition curves for positive values of curvature (using differenthorizontal and vertical scales) curve of linear curvature in violetline curve of polynomial curvature in green and curve of trigono-metric curvature in red

00 20 40 60 80 100 120

x (m)

y(m

)

minus40 minus20

minus1

minus2

minus3

minus4

minus5

minus6

minus7

Figure 8 Examples of geometric systems consisting of two circulararcs with consistent curvature (brown line) connected with differenttransition curves for negative values of curvature (using differenthorizontal and vertical scales) curve of linear curvature in violetline curve of polynomial curvature in green and curve of trigono-metric curvature in red

0

05

1

15

0 50 100 150x (m)

y(m

)

minus05

minus1

minus15

minus2

minus50

Figure 9 Examples of geometric systems consisting of two inversecircular arcs (brown line) connected with different types of transi-tion curves for positive values of input curvature (using differenthorizontal and vertical scales) curve of linear curvature in violetline curve of polynomial curvature in green and curve of trigono-metric curvature in red

Figures 9 and 10 illustrate geometric systems made up oftwo inverse circular arcs connected with various types oftransition curves

Here the layout of the output circular arc is much morediversifiedThe position of the arc obtained by the applicationof the transition curve of linear curvature diverts by severaldozenmeters from situationswhere curves of polynomial andtrigonometric curvature have been used The difference forcurves of polynomial and trigonometric curvature attains inthe numerical example analyzed an order of several meters

The presented examples are taken in the very nature ofthings at random They reveal wide possibilities for the pre-sented analytical method in modelling geometric systems Itsapplication makes it possible to generate diversified layout ofthe communication routes adapted to the field conditions andother circumstances Simultaneously the method ensurescorrectness of the obtained solution in view of the kinematicparameters magnitudes in force

12 Mathematical Problems in Engineering

0

05

1

15

2

0 50 100 150x (m)

y(m

)

minus05

minus1

minus15

minus50

Figure 10 Examples of geometric systems consisting of twoinverse circular arcs (brown line) connected with different typesof transition curves for negative values of input curvature (usingdifferent horizontal and vertical scales) curve of linear curvaturein violet line curve of polynomial curvature in green and curve oftrigonometric curvature in red

7 Conclusions

The presented universal method of modelling geometricalsystem of a communication route is based on the determina-tion of an adequate curvature equation by using differentialequationsThe technique enables us to design connections ofvarious geometrical elements of the route indicating diversifi-cation in its curvature It is characterized by an analytical wayof recording certain functions and moreover it is possible tospecify in advance the class of these functions

The sought-after connection is recorded in the form ofparametric equations 119909(119897) and 119910(119897) where parameter 119897 isthe position of a given point along the length of the curveThe determination of 119909(119897) and 119910(119897) needs the expansion ofintegrand functions into Maclaurin series and with regard toinverse curvatures into Taylor series

The determination of coordinates of an adequate tran-sition curve in rectangular coordinate system 119909 119910 makesit possible for an easy transfer of it to Gauss-Kruger (119883 119884)

conformal coordinates and its subsequent layout in fieldmaking use of the satellite measuring technique GNSS

The method can be applied to identify various typesof transition curves The paper gives examples of solvingthe problem involving linear change of curvature and thecurvature in polynomial and trigonometric form

The performed comparative analysis reveals a wide rangeof possibilities which are offered by the presented designmethod Its application enables us to generate diversifiedlayout of the route It simultaneously ensures a completecontrol of correctness of the obtained solutions in view ofsatisfying the geometric and kinematic conditions

The presented method of modelling curvature has auniversal character It can be used for vehicular roads andrailway lines as well In the case of railway lines therearises an additional possibility of modelling in a similar waythe superelevation ramp understood here as a determineddifference in the height of the rail courses

Conflict of Interests

The author declares that he has no conflict of interestsregarding the publication of this paper

References

[1] W Koc and C Specht ldquoApplication of the Polish active GNSSgeodetic network for surveying and design of the railroadrdquo inProceedings of the 1st International Conference on Road and RailInfrastructure (CETRA rsquo10) pp 757ndash762 May 2010

[2] P Lipar M Lakner T Maher andM Zura ldquoEstimation of roadcenterline curvature from raw GPS datardquo Baltic Journal of Roadand Bridge Engineering vol 6 no 3 pp 163ndash168 2011

[3] L Tasci and N Kuloglu ldquoInvestigation of a new transitioncurverdquoTheBaltic Journal of Road and Bridge Engineering vol 6no 1 pp 23ndash29 2011

[4] H-H Cai and G-J Wang ldquoA new method in highway routedesign joining circular arcs by a single C-Bezier curve withshape parameterrdquo Journal of Zhejiang University Science A vol10 no 4 pp 562ndash569 2009

[5] S Dimulyo Z Habib and M Sakai ldquoFair cubic transition bet-ween two circles with one circle inside or tangent to the otherrdquoNumerical Algorithms vol 51 no 4 pp 461ndash476 2009

[6] G Farin ldquoClass A Bezier curvesrdquo Computer Aided GeometricDesign vol 23 no 7 pp 573ndash581 2006

[7] Z Habib and M Sakai ldquoG2 pythagorean hodograph quintictransition between two circles with shape controlrdquo ComputerAided Geometric Design vol 24 no 5 pp 252ndash266 2007

[8] G Harary and A Tal ldquoThe natural 3D spiralrdquo Computer Graph-ics Forum vol 30 no 2 pp 237ndash246 2011

[9] K T Miura ldquoA general equation of aesthetic curves and its self-affinityrdquo Computer-Aided Design and Applications vol 3 no 1ndash4 pp 457ndash464 2006

[10] S H Yahaya M S Salleh and J M Ali ldquoSpur gear design withan S-shaped transition curve application using mathematicaand CAD toolsrdquo in Proceedings of the International Conferenceon Computer Technology and Development (ICCTD rsquo09) pp273ndash276 Dubai United Arab Emirates November 2009

[11] N Yoshida and T Saito ldquoQuasi-aesthetic curves in rationalcubic Bezier formsrdquo Computer-Aided Design and Applicationsvol 4 no 1ndash6 pp 477ndash486 2007

[12] R Ziatdinov N Yoshida and T-W Kim ldquoAnalytic parametricequations of log-aesthetic curves in terms of incomplete gammafunctionsrdquo Computer Aided Geometric Design vol 29 no 2 pp129ndash140 2012

[13] R J Grabowski ldquoSmooth curvilinear transitions along vehicu-lar roads and railway linesrdquo Science Papers of AGH Universityof Science and Technology no 82 Cracow Poland 1984(Polish)

[14] L T Klauder ldquoTrack transition curve geometry based on gegen-bauer polynomialsrdquo in Proceedings of the 6th International Con-ference on Railway Engineering May 2003

[15] W Koc Elements of Design Theory on Track Systems GdanskUniversity of Technology Gdansk Poland 2004 (Polish)

[16] E Mieloszyk and W Koc ldquoGeneral dynamic method for det-ermining transition curve equationsrdquo Rail InternationalmdashSchienen der Welt vol 10 pp 32ndash40 1991

[17] B Kufver ldquoRealigning railway in track renewalsmdashlinear versusS-shaped superelevation rampsrdquo in Proceedings of the 2ndInternational Conference on Railway Engineering May 1999

[18] W Koc and E Mieloszyk ldquoComparative analysis of selectedtransition curves by the use of dynamicmodelrdquoArchives of CivilEngineering vol 33 no 2 pp 239ndash261 1987 (Polish)

[19] S Gleason and D Gebre-Egziabher GNSS Applications andMethods Artech House Publishers Norwood NJ USA 2009

Mathematical Problems in Engineering 13

[20] B Hoffmann-Wellenhof H Lichtenegger and E Wasle GNSSGlobal Navigation Satellite SystemsmdashGPS GLONASS Galileo ampMore Springer Wien Austria 2008

[21] P Misra and P Enge Global Positioning System Signals Mea-surements and Performance Ganga-Jamuna Press LincolnMass USA 2012

[22] C Specht ldquoAccuracy and coverage of the modernized PolishMaritime differential GPS systemrdquo Advances in Space Researchvol 47 no 2 pp 221ndash228 2011

[23] W Koc and K Palikowska ldquoDynamic properties evaluation ofthe selected methods of joining route segments with differentcurvaturerdquo Technika Transportu Szynowego vol 9 pp 1785ndash1807 2012 (Polish)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

Table 1 Comparison of values 119909119870 119910119870 and 119904

119870for numerical examples analyzed

Curve 119897119896[m] Θ(119897

119896) [rad] 119904

119870119909119870[m] 119910

119870[m]

Case 1 1198961= 11200 radm and 119896

2= 1700 radm

Curve C0 40 0045238 0045269 39988 0825Curve C1 (p) 60 0067857 0067961 59963 1821Curve C1 (t) 62832 0071060 0071180 62796 1940

Case 2 1198961= 11200 radm and 119896

2= minus1700 radm

Curve C0 60 minus0017857 minus0017859 59999 0143Curve C1 (p) 90 minus0026786 minus0026792 89991 0627Curve C1 (t) 94248 minus0028050 minus0028057 94238 0714

Case 3 1198961= minus11200 radm and 119896

2= 1700 radm

Curve C0 60 001786 0017859 59999 minus0143Curve C1 (p) 90 0026786 0026792 89991 minus0627Curve C1 (t) 94248 0028050 0028057 94238 minus0714

Case 4 1198961= minus11200 radm and 119896

2= minus1700 radm

Curve C0 40 minus0045238 minus0045269 39988 minus0825Curve C1 (p) 60 minus0067857 minus0067961 59969 minus1821Curve C1 (t) 62832 minus0071060 minus0071180 62976 minus1940

0

1

2

3

4

5

6

0 20 40 60 80 100 120x (m)

y(m

)

minus40 minus20

Figure 7 Examples of geometric systems consisting of two circulararcs with consistent curvature (brown line) connected with differenttransition curves for positive values of curvature (using differenthorizontal and vertical scales) curve of linear curvature in violetline curve of polynomial curvature in green and curve of trigono-metric curvature in red

00 20 40 60 80 100 120

x (m)

y(m

)

minus40 minus20

minus1

minus2

minus3

minus4

minus5

minus6

minus7

Figure 8 Examples of geometric systems consisting of two circulararcs with consistent curvature (brown line) connected with differenttransition curves for negative values of curvature (using differenthorizontal and vertical scales) curve of linear curvature in violetline curve of polynomial curvature in green and curve of trigono-metric curvature in red

0

05

1

15

0 50 100 150x (m)

y(m

)

minus05

minus1

minus15

minus2

minus50

Figure 9 Examples of geometric systems consisting of two inversecircular arcs (brown line) connected with different types of transi-tion curves for positive values of input curvature (using differenthorizontal and vertical scales) curve of linear curvature in violetline curve of polynomial curvature in green and curve of trigono-metric curvature in red

Figures 9 and 10 illustrate geometric systems made up oftwo inverse circular arcs connected with various types oftransition curves

Here the layout of the output circular arc is much morediversifiedThe position of the arc obtained by the applicationof the transition curve of linear curvature diverts by severaldozenmeters from situationswhere curves of polynomial andtrigonometric curvature have been used The difference forcurves of polynomial and trigonometric curvature attains inthe numerical example analyzed an order of several meters

The presented examples are taken in the very nature ofthings at random They reveal wide possibilities for the pre-sented analytical method in modelling geometric systems Itsapplication makes it possible to generate diversified layout ofthe communication routes adapted to the field conditions andother circumstances Simultaneously the method ensurescorrectness of the obtained solution in view of the kinematicparameters magnitudes in force

12 Mathematical Problems in Engineering

0

05

1

15

2

0 50 100 150x (m)

y(m

)

minus05

minus1

minus15

minus50

Figure 10 Examples of geometric systems consisting of twoinverse circular arcs (brown line) connected with different typesof transition curves for negative values of input curvature (usingdifferent horizontal and vertical scales) curve of linear curvaturein violet line curve of polynomial curvature in green and curve oftrigonometric curvature in red

7 Conclusions

The presented universal method of modelling geometricalsystem of a communication route is based on the determina-tion of an adequate curvature equation by using differentialequationsThe technique enables us to design connections ofvarious geometrical elements of the route indicating diversifi-cation in its curvature It is characterized by an analytical wayof recording certain functions and moreover it is possible tospecify in advance the class of these functions

The sought-after connection is recorded in the form ofparametric equations 119909(119897) and 119910(119897) where parameter 119897 isthe position of a given point along the length of the curveThe determination of 119909(119897) and 119910(119897) needs the expansion ofintegrand functions into Maclaurin series and with regard toinverse curvatures into Taylor series

The determination of coordinates of an adequate tran-sition curve in rectangular coordinate system 119909 119910 makesit possible for an easy transfer of it to Gauss-Kruger (119883 119884)

conformal coordinates and its subsequent layout in fieldmaking use of the satellite measuring technique GNSS

The method can be applied to identify various typesof transition curves The paper gives examples of solvingthe problem involving linear change of curvature and thecurvature in polynomial and trigonometric form

The performed comparative analysis reveals a wide rangeof possibilities which are offered by the presented designmethod Its application enables us to generate diversifiedlayout of the route It simultaneously ensures a completecontrol of correctness of the obtained solutions in view ofsatisfying the geometric and kinematic conditions

The presented method of modelling curvature has auniversal character It can be used for vehicular roads andrailway lines as well In the case of railway lines therearises an additional possibility of modelling in a similar waythe superelevation ramp understood here as a determineddifference in the height of the rail courses

Conflict of Interests

The author declares that he has no conflict of interestsregarding the publication of this paper

References

[1] W Koc and C Specht ldquoApplication of the Polish active GNSSgeodetic network for surveying and design of the railroadrdquo inProceedings of the 1st International Conference on Road and RailInfrastructure (CETRA rsquo10) pp 757ndash762 May 2010

[2] P Lipar M Lakner T Maher andM Zura ldquoEstimation of roadcenterline curvature from raw GPS datardquo Baltic Journal of Roadand Bridge Engineering vol 6 no 3 pp 163ndash168 2011

[3] L Tasci and N Kuloglu ldquoInvestigation of a new transitioncurverdquoTheBaltic Journal of Road and Bridge Engineering vol 6no 1 pp 23ndash29 2011

[4] H-H Cai and G-J Wang ldquoA new method in highway routedesign joining circular arcs by a single C-Bezier curve withshape parameterrdquo Journal of Zhejiang University Science A vol10 no 4 pp 562ndash569 2009

[5] S Dimulyo Z Habib and M Sakai ldquoFair cubic transition bet-ween two circles with one circle inside or tangent to the otherrdquoNumerical Algorithms vol 51 no 4 pp 461ndash476 2009

[6] G Farin ldquoClass A Bezier curvesrdquo Computer Aided GeometricDesign vol 23 no 7 pp 573ndash581 2006

[7] Z Habib and M Sakai ldquoG2 pythagorean hodograph quintictransition between two circles with shape controlrdquo ComputerAided Geometric Design vol 24 no 5 pp 252ndash266 2007

[8] G Harary and A Tal ldquoThe natural 3D spiralrdquo Computer Graph-ics Forum vol 30 no 2 pp 237ndash246 2011

[9] K T Miura ldquoA general equation of aesthetic curves and its self-affinityrdquo Computer-Aided Design and Applications vol 3 no 1ndash4 pp 457ndash464 2006

[10] S H Yahaya M S Salleh and J M Ali ldquoSpur gear design withan S-shaped transition curve application using mathematicaand CAD toolsrdquo in Proceedings of the International Conferenceon Computer Technology and Development (ICCTD rsquo09) pp273ndash276 Dubai United Arab Emirates November 2009

[11] N Yoshida and T Saito ldquoQuasi-aesthetic curves in rationalcubic Bezier formsrdquo Computer-Aided Design and Applicationsvol 4 no 1ndash6 pp 477ndash486 2007

[12] R Ziatdinov N Yoshida and T-W Kim ldquoAnalytic parametricequations of log-aesthetic curves in terms of incomplete gammafunctionsrdquo Computer Aided Geometric Design vol 29 no 2 pp129ndash140 2012

[13] R J Grabowski ldquoSmooth curvilinear transitions along vehicu-lar roads and railway linesrdquo Science Papers of AGH Universityof Science and Technology no 82 Cracow Poland 1984(Polish)

[14] L T Klauder ldquoTrack transition curve geometry based on gegen-bauer polynomialsrdquo in Proceedings of the 6th International Con-ference on Railway Engineering May 2003

[15] W Koc Elements of Design Theory on Track Systems GdanskUniversity of Technology Gdansk Poland 2004 (Polish)

[16] E Mieloszyk and W Koc ldquoGeneral dynamic method for det-ermining transition curve equationsrdquo Rail InternationalmdashSchienen der Welt vol 10 pp 32ndash40 1991

[17] B Kufver ldquoRealigning railway in track renewalsmdashlinear versusS-shaped superelevation rampsrdquo in Proceedings of the 2ndInternational Conference on Railway Engineering May 1999

[18] W Koc and E Mieloszyk ldquoComparative analysis of selectedtransition curves by the use of dynamicmodelrdquoArchives of CivilEngineering vol 33 no 2 pp 239ndash261 1987 (Polish)

[19] S Gleason and D Gebre-Egziabher GNSS Applications andMethods Artech House Publishers Norwood NJ USA 2009

Mathematical Problems in Engineering 13

[20] B Hoffmann-Wellenhof H Lichtenegger and E Wasle GNSSGlobal Navigation Satellite SystemsmdashGPS GLONASS Galileo ampMore Springer Wien Austria 2008

[21] P Misra and P Enge Global Positioning System Signals Mea-surements and Performance Ganga-Jamuna Press LincolnMass USA 2012

[22] C Specht ldquoAccuracy and coverage of the modernized PolishMaritime differential GPS systemrdquo Advances in Space Researchvol 47 no 2 pp 221ndash228 2011

[23] W Koc and K Palikowska ldquoDynamic properties evaluation ofthe selected methods of joining route segments with differentcurvaturerdquo Technika Transportu Szynowego vol 9 pp 1785ndash1807 2012 (Polish)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

0

05

1

15

2

0 50 100 150x (m)

y(m

)

minus05

minus1

minus15

minus50

Figure 10 Examples of geometric systems consisting of twoinverse circular arcs (brown line) connected with different typesof transition curves for negative values of input curvature (usingdifferent horizontal and vertical scales) curve of linear curvaturein violet line curve of polynomial curvature in green and curve oftrigonometric curvature in red

7 Conclusions

The presented universal method of modelling geometricalsystem of a communication route is based on the determina-tion of an adequate curvature equation by using differentialequationsThe technique enables us to design connections ofvarious geometrical elements of the route indicating diversifi-cation in its curvature It is characterized by an analytical wayof recording certain functions and moreover it is possible tospecify in advance the class of these functions

The sought-after connection is recorded in the form ofparametric equations 119909(119897) and 119910(119897) where parameter 119897 isthe position of a given point along the length of the curveThe determination of 119909(119897) and 119910(119897) needs the expansion ofintegrand functions into Maclaurin series and with regard toinverse curvatures into Taylor series

The determination of coordinates of an adequate tran-sition curve in rectangular coordinate system 119909 119910 makesit possible for an easy transfer of it to Gauss-Kruger (119883 119884)

conformal coordinates and its subsequent layout in fieldmaking use of the satellite measuring technique GNSS

The method can be applied to identify various typesof transition curves The paper gives examples of solvingthe problem involving linear change of curvature and thecurvature in polynomial and trigonometric form

The performed comparative analysis reveals a wide rangeof possibilities which are offered by the presented designmethod Its application enables us to generate diversifiedlayout of the route It simultaneously ensures a completecontrol of correctness of the obtained solutions in view ofsatisfying the geometric and kinematic conditions

The presented method of modelling curvature has auniversal character It can be used for vehicular roads andrailway lines as well In the case of railway lines therearises an additional possibility of modelling in a similar waythe superelevation ramp understood here as a determineddifference in the height of the rail courses

Conflict of Interests

The author declares that he has no conflict of interestsregarding the publication of this paper

References

[1] W Koc and C Specht ldquoApplication of the Polish active GNSSgeodetic network for surveying and design of the railroadrdquo inProceedings of the 1st International Conference on Road and RailInfrastructure (CETRA rsquo10) pp 757ndash762 May 2010

[2] P Lipar M Lakner T Maher andM Zura ldquoEstimation of roadcenterline curvature from raw GPS datardquo Baltic Journal of Roadand Bridge Engineering vol 6 no 3 pp 163ndash168 2011

[3] L Tasci and N Kuloglu ldquoInvestigation of a new transitioncurverdquoTheBaltic Journal of Road and Bridge Engineering vol 6no 1 pp 23ndash29 2011

[4] H-H Cai and G-J Wang ldquoA new method in highway routedesign joining circular arcs by a single C-Bezier curve withshape parameterrdquo Journal of Zhejiang University Science A vol10 no 4 pp 562ndash569 2009

[5] S Dimulyo Z Habib and M Sakai ldquoFair cubic transition bet-ween two circles with one circle inside or tangent to the otherrdquoNumerical Algorithms vol 51 no 4 pp 461ndash476 2009

[6] G Farin ldquoClass A Bezier curvesrdquo Computer Aided GeometricDesign vol 23 no 7 pp 573ndash581 2006

[7] Z Habib and M Sakai ldquoG2 pythagorean hodograph quintictransition between two circles with shape controlrdquo ComputerAided Geometric Design vol 24 no 5 pp 252ndash266 2007

[8] G Harary and A Tal ldquoThe natural 3D spiralrdquo Computer Graph-ics Forum vol 30 no 2 pp 237ndash246 2011

[9] K T Miura ldquoA general equation of aesthetic curves and its self-affinityrdquo Computer-Aided Design and Applications vol 3 no 1ndash4 pp 457ndash464 2006

[10] S H Yahaya M S Salleh and J M Ali ldquoSpur gear design withan S-shaped transition curve application using mathematicaand CAD toolsrdquo in Proceedings of the International Conferenceon Computer Technology and Development (ICCTD rsquo09) pp273ndash276 Dubai United Arab Emirates November 2009

[11] N Yoshida and T Saito ldquoQuasi-aesthetic curves in rationalcubic Bezier formsrdquo Computer-Aided Design and Applicationsvol 4 no 1ndash6 pp 477ndash486 2007

[12] R Ziatdinov N Yoshida and T-W Kim ldquoAnalytic parametricequations of log-aesthetic curves in terms of incomplete gammafunctionsrdquo Computer Aided Geometric Design vol 29 no 2 pp129ndash140 2012

[13] R J Grabowski ldquoSmooth curvilinear transitions along vehicu-lar roads and railway linesrdquo Science Papers of AGH Universityof Science and Technology no 82 Cracow Poland 1984(Polish)

[14] L T Klauder ldquoTrack transition curve geometry based on gegen-bauer polynomialsrdquo in Proceedings of the 6th International Con-ference on Railway Engineering May 2003

[15] W Koc Elements of Design Theory on Track Systems GdanskUniversity of Technology Gdansk Poland 2004 (Polish)

[16] E Mieloszyk and W Koc ldquoGeneral dynamic method for det-ermining transition curve equationsrdquo Rail InternationalmdashSchienen der Welt vol 10 pp 32ndash40 1991

[17] B Kufver ldquoRealigning railway in track renewalsmdashlinear versusS-shaped superelevation rampsrdquo in Proceedings of the 2ndInternational Conference on Railway Engineering May 1999

[18] W Koc and E Mieloszyk ldquoComparative analysis of selectedtransition curves by the use of dynamicmodelrdquoArchives of CivilEngineering vol 33 no 2 pp 239ndash261 1987 (Polish)

[19] S Gleason and D Gebre-Egziabher GNSS Applications andMethods Artech House Publishers Norwood NJ USA 2009

Mathematical Problems in Engineering 13

[20] B Hoffmann-Wellenhof H Lichtenegger and E Wasle GNSSGlobal Navigation Satellite SystemsmdashGPS GLONASS Galileo ampMore Springer Wien Austria 2008

[21] P Misra and P Enge Global Positioning System Signals Mea-surements and Performance Ganga-Jamuna Press LincolnMass USA 2012

[22] C Specht ldquoAccuracy and coverage of the modernized PolishMaritime differential GPS systemrdquo Advances in Space Researchvol 47 no 2 pp 221ndash228 2011

[23] W Koc and K Palikowska ldquoDynamic properties evaluation ofthe selected methods of joining route segments with differentcurvaturerdquo Technika Transportu Szynowego vol 9 pp 1785ndash1807 2012 (Polish)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 13

[20] B Hoffmann-Wellenhof H Lichtenegger and E Wasle GNSSGlobal Navigation Satellite SystemsmdashGPS GLONASS Galileo ampMore Springer Wien Austria 2008

[21] P Misra and P Enge Global Positioning System Signals Mea-surements and Performance Ganga-Jamuna Press LincolnMass USA 2012

[22] C Specht ldquoAccuracy and coverage of the modernized PolishMaritime differential GPS systemrdquo Advances in Space Researchvol 47 no 2 pp 221ndash228 2011

[23] W Koc and K Palikowska ldquoDynamic properties evaluation ofthe selected methods of joining route segments with differentcurvaturerdquo Technika Transportu Szynowego vol 9 pp 1785ndash1807 2012 (Polish)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of