selariu supermathematics functions, editor florentin smarandache

8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

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Techno-Art of Selariu SuperMathematics Functions

A e r o n a u t i c s C a p s u l e

2007


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Techno-Art of Selariu SuperMathematics Functions

Editor: Florentin Smarandache

ARP2007


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Front cover art image, called Aeronautics Capsule, by Mircea Eugen elariu.

This book can be ordered in a paper bound reprint from:

Books on DemandProQuest Information & Learning

(University of Microfilm International)

300 N. Zeeb RoadP.O. Box 1346, Ann Arbor

MI 48106-1346, USA

Tel.: 1-800-521-0600 (Customer Service)http://wwwlib.umi.com/bod/basic

Copyright 2007 by American Research Press, Editor and the Authors for their SuperMathematics

Functions and Images.

Many books can be downloaded from the following

Digital Library of Arts & Letters:

http://www.gallup.unm.edu/~smarandache/eBooksLiterature.htm

Peer Reviewers:1. Prof. Mihly Bencze, University of Braov, Romania.

2. Prof. Valentin Boju, Ph. D.

Officer of the Order Cultural Merit, Category Scientific ResearchMontrealTech - Institut de Technologie de Montral

Director, MontrealTech Press

P. O. Box 78574, Station Wilderton

Montral, Qubec, H3S 2W9, Canada

(ISBN-10): 1-59973-037-5

(ISBN-13): 978-1-59973-037-0

(EAN): 9781599730370

Printed in the United States of America


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C-o-n-t-e-n-t-s

A e r o n a u t i c s c a p s u l e first cover

FOREWORD (FOR SUPERMATHEMATICS FUNCTIONS) 6-16

Selariu SuperMathematics Functions

& Other SuperMathematics Functions 17

D O U B L E C L E P S Y D R A 18S U P E R M A T H E M A T I C S F L O W E R S

19The Ballet of the Functions 20

J a c u z z i 21D a m a g e d p a r t o f T I T A N I C 22K A Z A T C I O K (Russian Popular Dance) 23F l y i n g B i r d 1 24F l y i n g B i r d 2 25M U L T I C O L O R E D S U N 26R e d S u n 27T h e d o u b l e N o z z l e f o r N A S A 28T h e s u p e r m a t h e m a t i c s C o m e t 29The Lake of Swans 30

+ The Dance of Swords 30+ The Nut Cracker 30+ The Decease of Swan 30

T h e F l o w e r i n g 31The supermathematics ring surface 32The ex-centric sphere 33T h e s u p e r m a t h e m a t i c s Screw Surface 34T h e T r o j a n H o r s e 35T h e A m p h o r a s 36B U D D H A 37J E T P L A N E 38T R O U B L E D L A N D

o r D O U B L E A N A L Y T I C A L E X C E N T R I CF U N C T I O N 39

S E L F - P I E R C E B O D Y 40H I L L S a n d V A L L E Y S 41B e r n o u l l i s L e m n i s c a t e, C a s s i n i s O v a l s a n d o t h e r s 42Continuous transforming of a circle into a haystack 43C y c l i c a l S y m m e t r y 44S m a r a n d a c h e S t e p p e d F u n c t i o n s 45S C R I B B L I N G S W I T H . . . H E A D A N D T A L E 46Q U A D R I P O D 47E X C E N T R I C S Y M M E T R Y 48


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D R A C U L A S C A S T L E 49T U N I N G F O R K 50

d e x O I D S + r e x O I D S 51E x c e n t r i c g e o m e t r y , p r i s m a t i c s o l i d s 52V a s e 53H E X A G O N A L T O R U S 54O p e n s q u a r e t o r u s 55D o u b l e s q u a r e t o r u s 56S i n u o u s C o r r u g a t e W a s h e r s

o r N a n o - P e r i s t a l t i c E n g i n e 57I G L O O + T h e m a g i c c a r p e t 58Multiple Ex Centric Circular SuperMathematics Functions 59E X C E N T R I C T O R U S R I N G 60H Y P E R S O N I C J E T A I R P L A N E 61P L U M P V A S E 62A R R O W S 63H Y P E R B O L I C Q U A D R A T I C C Y L I N D E R 1 64H Y P E R B O L I C Q U A D R A T I C C Y L I N D E R 2 65E X - C E N T R I C F U L L S P R I N G 66E X - C E N T R I C E M P T Y S P R I N G 67E X C E N T R I C P E N T A G O N H E L I X 68C Y L I N D E R S w i t h C O L L A R S 69U n i c u r s a l S u p e r m a t h e m a t i c s F u n c t i o n s 1

+ O l d w o m a n f r o m C a r p a t i M o u n t a i n ( R o m a ni a ) 70U n i c u r s a l S u p e r m a t h e m a t i c s F u n c t i o n s 2 71U n i c u r s a l Su p e r m a t h e m a t i c s F u n c t i o n s 3

+ W a l k i n g P i n g u i n s 72T o D o u b l e a n d S i m p l e C a n o e 73R o m a n i a n f o l k d a n c e 74P i l l o w 75T e r r a w i t h M a r k e d M e r i d i a n s 76S u p e r m a t h e m a t i c s C o l u m n s 77A e r o d y n a m i c S o l i d 78H a l v i n g C u r v e 79The crook lines (s 0) - a generalization of straight lines ( s = 0 ) 80-82A R A B E S Q U E S 1 83S T A R S 84

H y s t e r e t i c C u r v e s 1 85H y s t e r e t i c C u r v e s 2 86H y s t e r e t i c C u r v e s 3 87E x C e n t r i c C i r c u l a r C u r v e s w i t h E x C e n t r i c V a r i a b l e 88F i l i g r e e 1 89F I L I G R E E 2 90U s h a p e d C u r v e s i n 2 D a n d 3 D 91E x p l o s i o n s 92S p a t i a l F i g u r e 93P l a n e t s a n d s t a r s 94E x C e n t r i c C i r c l e ( n = 1 ) a n d A s t e r o i d ( n = 2, 4 , 6 ) 95E x C e n t r i c A s t e r o i d ( n = 3, 5 ,7 ,9) 96E x C e n t r i c L e m n i s c a t e s 97B u t t e r f l y w i t h S y m m e t r i c a l C e n t e r 1 98B u t t e r f l y w i t h S y m m e t r i c a l C e n t e r 2 99B u t t e r f l y w i t h S y m m e t r i c a l C e n t e r 3 100B u t t e r f l y R a p i d l y F l a p p i n g t h e W i n g s 101F l o w e r w i t h F o u r P e t a l e s 102


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E c C e n t r i c P y r a m i d 103A e r o d y n a m i c P r o f i l e

w i t h S u p e r m a t h e m a t i c s F u n c t i o n s 1 104A e r o d y n a m i c P r o f i l e

w i t h S u p e r m a t h e m a t i c s F u n c t i o n s 2 105S A L T C e l l a r 106S U P E R M A T H E M A T I C S T O W E R 107THE CONTINUOUS TRANSFORMATION OF A RIGHT TRIANGLE INTO

ITS HYPOTENUSE

+ THE CONTINUOUS TRANSFORMATION OF QUADRATE (Rx = Ry)

OR RECTANGLE (Rx Ry) INTO ITS DIAGONAL 108Mo d i f i e d c e x a n d s e x 109S i n u o u s S u r f a c e w i t h A n a l i t i c a l S u p e r m a t h e m a t i c s

F u n c t i o n s 110S u p e r m a t h e m a t i c s S p i r a l

+ S u p e r m a t h e m a t i c s P a r a b l e s 111A R A B E S Q U E S 2 112S u p e r m a t h e m a t i c s f u n c t i o n s cex xy and sex xy 113S u p e r m a t h e m a t i c s f u n c t i o n s rex xy and dex xy 114E x C e n t r i c F o l k l o r e C a r p e t 1 115E x C e n t r i c F o l k l o r e C a r p e t 2 116E x C e n t r i c F o l k l o r e C a r p e t 3 117W A T E R F A L L I N G 118

S I N G L E and D O U B L E K C Y L I N D E R 119S U P E R M A T H E M A T I C A L K N O T S H A P E D B R E A D

a n d O N E C R A C K N E L ( P R E T Z E L) 120S i x C o n o p y r a m i d s 121F O U R C O N O P Y R A M I D S V I E W E D F R O M A B O V E 122D O U B L E C O N O P Y R A M I D

or the transformation of circle into a square with circular ex-centric supermathematics function dex orquadrilobic functions cosq and sinq v 123

E X - C E N T R I C S a n d V A L E R I U A L A C I C U A D R O L O B S 124P E R F E C T C U B E 125E x c e n t r i c c i r c u l a r c u r v e s 1 126E x c e n t r i c c i r c u l a r c u r v e s 2

o r E x c e n t r i c L i s s a j o u s c u r v e s 127

References in SuperMathematics 128Postface back cover


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FOREWORD

(FOR SUPER-MATHEMATICS FUNCTIONS)

In this album we include the so called Super-Mathematics functions (SMF), which

constitute the base for, most often, generating, technical, neo-geometrical objects, therefore lessartistic.These functions are the results of 38 years of research, which began at University of Stuttgart in

1969. Since then, 42 related works have been published, written by over 19 authors, as shown in theReferences.

The name was given by the regretted mathematician Professor Emeritus Doctor Engineer

Gheorghe Silas who, at the presentation of the very first work in this domain, during the First National Conference of Vibrations in Machine Constructions, Timioara, Romania, 1978, namedCIRCULAR EX-CENTRIC FUNCTIONS, declared: Young man, you just discovered not only

some functions, but a new mathematics, a supermathematics! I was glad, at my age of 40, like ateenager. And I proudly found that he might be right!

The prefix super is justified today, to point out the birth of the new complements inmathematics, joined together under the name of Ex-centric Mathematics (EM), with much more

important and infinitely more numerous entities than the existing ones in the actual mathematics,which weare obliged to call it Centric Mathematics (CM.)

To each entity from CM corresponds an infinity of similar entities in EM, therefore the

Supermathematics (SM) is the reunion of the two domains: SM = CM EM, where CM is aparticular case of null ex-centricity ofEM. Namely, CM = SM(e = 0). To each known function in

CM corresponds an infinite family of functions in EM, and in addition, a series of new functions

appear, with a wide range of applications in mathematics and technology.In this way, to x = coscorresponds the family of functions x = cex = cex (, s, ) where s =

e/R and are the polar coordinates of the ex-center S(s,), which corresponds to theunity/trigonometric circle orE(e, ),which corresponds to a certain circle of radius R, considered aspoleof a straight line d, which rotates around EorS with the position angle , generating in this waythe ex-centric trigonometric functions, or ex-centric circular supermathematics functions (EC-SMF),by intersecting d with the unity circle (see.Fig.1). Amongst them the ex-centriccosineof, denotedcex = x, where x is the projection of the point W, which is the intersection of the straight line withthe trigonometric circle C(1,O), or the Cartesian coordinates of the point W. Because a straight line,

passing through S, interior to the circle (s 1e < R), intersects the circle in two points W1 and W2,which can be denoted W1,2, it results that there are two determinations of the ex-centric circular

supermathematics functions (EC-SMF): a principal one of index 1cex1, and a secondary one cex2,of index 2, denoted cex1,2.EandS were named ex-centrebecause they were excluded from thecenter O(0,0). This exclusion leads to the apparition of EM and implicitly of SM. By this, the

number of mathematical objects grew from one to infinity: to a unique function from CM, for

example cos, corresponds aninfinity of functionscex, due to the possibilities of placing theex-center Sand/orE in the plane.

S(e, ) can take an infinite number of positions in the plane containing the unity ortrigonometric circle. For each position ofS and E we obtain a functioncex. IfS is a fixed point,then we obtain the ex-centric circular SM functions (EC-SMF), with fixed ex-center, or with constant

s and . But S or E can take different positions, in the plane, by various rules or laws, while the


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straight line which generates the functions by its intersection with the circle, rotates with the angle aroundSandE.

Fig.1 Definition of Ex-Centric Circular Supermathematics Functions (EC-SMF)

In the last case, we have an EC-SMF of ex-center variable point S/E, which means s = s ()and/or = (). If the variable position ofS/E is represented also by EC-SMF of the same ex-centerS(s, ) or by another ex-center S1[s1 = s1(), 1 = 1 ()], then we obtain functions of double ex-centricity. By extrapolation, well obtain functions of triple, and multiple ex-centricity. Therefore,

EC-SMFare functions of as many variables as we want or as many as we need .

If the distances from O to the points W1,2on the circle C(1,O) areconstant and equal to the

radius R = 1 of the trigonometric circle C, distances that will be named ex-centric radiuses, the

distances from S to W1,2 denoted by r1,2 are variable and are named ex-centric radiuses of the unitycircle C(1,O) and represent, in the same time, new ex-centric circular supermathematics functions

(EC-SMF), which were named ex-centric radial functions, denoted rex1,2 , if are expressed infunction of the variablenamed ex-centric and motor, which is the angle from the ex-centerE. Or,denoted Rex1,2, if it is expressed in function of the angle or the centric variable, the angle atO(0,0). The W1,2 are seen under the angles 1,2from O(0,0) and under the angles and + fromS(e, ) and E. The straight line d is divided by S din the twosemi-straight lines, one positived +and the other negative d

. For this reason, we can considerr1 = rex1a positive oriented segment

on d ( r1 > 0) andr2= rex2a negative orientedsegmenton d(r2 < 0) in the negative sense ofthe semi-straight line d

.

M1

W1

x

y

O

S

E

cex1

sex1

cex2

sex2

OS = s OE = e

OW1 = OW2 = 1OM1 = OM2 = R

SW1 = r1 = rex1SW2 = r2 = rex2

EM1 = R.r1 = R.rex1

EM2 = R.r2 = R.rex2

W1OA = 1W2OA = 2SOA = S(s,)E(e,)

M1,2 (R, 1,2)

W1,2 (1, 1,2)

A

aex1,2 = 1,2 () = 1,2() =

bex1,2 == arcsin[s.sin(-)]cex1,2 = cos 1,2sex1,2 = sin 1,2

W2

dex1,2 =

d

d 2,1=

=1 -

)(sin1

)cos(.

22

s

s

Dex 1,2 =

21

d

d


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Using simple trigonometric relations, in certain triangles OEW1,2, or, more precisely, writing

the sine theorem (as function of) and Pitagoras generalized theorem (for the variables1,2) in thesetriangles, it immediately results the invariant expressions of the ex-centric radial functions:

r1,2 () = rex1,2= s.cos() )(sin122 s

and

r1,2 (1,2) = Rex1,2 = )cos(..212 + ss .

All EC-SMF have invariant expressions, and because of that they dont need to be tabulated,tabulated being only the centric functions from CM, which are used to express them. In all of their

expressions, we will always find one of the square roots of the previous expressions, of ex-centric

radial functions.Finding these two determinations is simple: for+ (plus) in front of the square roots we always

obtain the first determination (r1 > 0) and for the(minus) sign we obtain the second determination(r2 < 0). The rule remains true for allEC-SMF. By convention, the first determination, of index 1, can

be used or written without index.Some remarks about these REX (King) functions:

The ex-centric radial functions are the expression of the distance between two points,in the plane, in polar coordinates: S(s, ) and W1,2 (R=1, 1,2), on the direction of thestraight line d, skewedat an angle in relation to Ox axis;

Therefore, using exclusively these functions, we can express the equations of allknown plane curves, as well as of other new ones, which surfaced with theintroduction ofEM.An example is represented by Booths lemniscates (see Fig. 2, a,

b, c), expressed, in polar coordinates, by the equation:

() = R(rex1+rex 2) =2 s.Rcos( - ) forR=1, = 0 ands [0, 3].

-1 -0.5 0.5 1

-0.4

-0.2

0.2

0.4

-1 -0.5 0.5

-0.2

-0.1

0.1

0.2

Fig.2,a Booths Lemniscates for R = 1 and

numerical ex-centricity e [1.1, 2]Fig. 2,b BoothsLemniscates forR = 1and

numerical ex-centricity e [2.1, 3]

Another consequence is the generalization of the definition of a circle:The Circle is the plane curve whose points M are at the distances r() = R.rex =R.rex [, E(e, )] in relation to a certain point from the circles plane E(e, ).


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IfSO(0,0), thens = 0 and rex = 1 = constant, andr() = R= constant, we obtainthe circles classical definition: the points situated at the same distance R from apoint, the center of the circle.

Booth Lemniscate Functions

Polar coordinate equation with

supermathematics circle functions rex 1,2 : = R (rex1 + rex2 )for

circle radius R = 1

and

the numerical ex-centricity s [ 0, 1 ]

Fig. 2,c

The functionsrex and Rex expresses the transfer functions of zero degree, or ofthe position of transfer, from the mechanism theory, and it is the ratio between the


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parameter R(1,2), which positions the conducted element OM1,2 and parameterR.r1,2(), which positions the leader elementEM1,2.Between these two parameters, there are the following relations, which can be deduced

similarly easy from Fig. 1 that defines EC-SMF.

Between the position angles of the two elements, leaded and leader, there are the

following relations:1,2 = arcsin[e.sin()] = 1,2() =aex1,2and

= 1,2 1,2(1,2 )= 1,2 arcsin[)cos(..21

)sin(.

2,1

2

2,1

+

ss

s] = Aex (1,2 ).

The functions aex 1,2 and Aex 1,2 are EC-SMF, called ex-centric amplitude,because of their usage in defining the ex-centric cosine and sine from EC-SMF, in the

same manner as the amplitude function or amplitudinus am(k,u) is used for defining

the elliptical Jacobi functions:sn(k,u) = sn[am(k,u)], cn(k,u) = cos[am(k,u)],

or:

cex1,2 = cos(aex1,2) , Cex1,2 = cos(Aex1,2)and

sex 1,2 = sin (aex1,2), Sex 1,2 = cos (Aex 1,2)

The radial ex-centric functions can be considered as modules of the position vectors

2,1r for the W1,2 on the unity circle C (1,O). These vectors are expressed by the

following relations:

radrexr .2,12,1 =

,

where rad is the unity vector of variable direction, or the versor/phasor of thestraight line direction d

+, whose derivative is the phasor der = d(rad)/d and

represents normal vectors on the straight lines OW1,2, directions, tangent to the circle in

the W1,2. They are named the centric derivative phasors. In the same time, themodulus rad function is the corresponding, in CM, of the function rex fors = 0 = when rex = 1 andder 1,2 are the tangent versors to the unity circle in W1,2.

The derivative of the

2,1r vectors are the velocity vectors:

2,12,1

2,1

2,1 .

derdexd

rdv ==

of the W1,2 C points in their rotating motion on the circle, with velocities of variablemodulus v1,2 =dex1,2, when the generating straight line d rotates around the ex-centerS with a constant angular speed and equal to the unity, namely = 1.The velocityvectors have the expressions presented above, where der 1,2are the phasors of centricradiuses R1,2 of module 1 and of1,2 directions. The expressions of the functions EC-


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SMdex1,2 ,ex-centric derivativeof, are, in the same time, also the1,2() anglesderivatives, as function of the motor or independent variable , namely

dex1,2 = d1,2 ()/d = 1)(sin.1

)cos(.

22

s

s

as function of, and

Dex 1,2 = d()/d1,2 =2,1

2

2,1

2,1

2

2,1

Re

)cos(.1

)cos(..21

)cos(.1

x

s

ss

s =

+

,

as functions of 1,2 .It has been demonstrated that the ex-centric derivative functionsEC-SMexpress the

transfer functions of the first order, or of the angular velocity, from the MechanismsTheory, forall (!) known plane mechanisms.

The radial ex-centric function rex expresses exactly the movement of push-pullmechanism S = R. rex, whose motor connecting rod has the length r, equal with ethe real ex-centricity,and the length of the crank is equal to R, the radius of the circle,a very well-known mechanism, because it is a component of all automobiles, except

those with Wankel engine.The applications of radial ex-centric functions could continue, but we will concentrate now on

the more general applications ofEC-SMF.

Concretely, to the unique forms as those of the circle, square, parabola, ellipse, hyperbola,

different spirals, etc. from CM, which are now grouped under the name of centrics, correspond aninfinity ofex-centrics of the same type: circular, square (quadrilobe), parabolic, elliptic, hyperbolic,

various spirals ex-centrics, etc. Any ex-centric function, with null ex-centricity (e = 0), degenerates

into a centric function, which represents, at the same time its generating curve. Therefore, the CM

itself belongs to EM, for the unique case (s = e = 0), which is one case from an infinity of possiblecases, in which a point named eccenterE(e, )can be placed in plane. In this case, E is overleapingon one or two points named center: the origin O(0,0) of a frame, considered the origin O(0,0) of the

referential system, and/or the centerC(0,0) of the unity circle for circular functions, respectively, thesymmetry center of the two arms of the equilateral hyperbola, for hyperbolic functions.

It was enough that a point E be eliminated from the center (O and/orC) to generate from the

old CM a new world ofEM. The reunion of these two worlds gave birth to theSM world.This discovery occurred in the city of the Romanian Revolution from 1989, Timioara, which

is the same citywhere on November 3rd

, 1823 Janos Bolyai wrote: FromnothingIve created a new

world. With these words, he announced the discovery of the fundamental formula of the first non-

Euclidean geometry.

He from nothing, I in a joint effort, proliferated the periodical functions which are so

helpful to engineers to describe some periodical phenomena. In this way, I have enriched the

mathematics with new objects.

When Euler defined the trigonometric functions, as direct circular functions, if he wouldnthave chosen three superposed points: the origin O, the center of the circle C and S as a pole of a

semi straight line, with which he intersected the trigonometric/unity circle, the EC-SMF would havebeen discovered much earlier, eventually under another name.

Depending on the way of the split (we isolate one point at the time from the superposed

ones, or all of them at once), we obtain the following types ofSMF:

O C S Centric functions belonging to CM;


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and those which belong toEM are:

O C S Ex-centric Circular SupermathematicsFunctions (EC-SMF);O C S Elevated Circular SupermathematicsFunctions (ELC-SMF);O C S Exotic Circular SupermathematicsFunctions (EXC-SMF).These new mathematics complements, joined under the temporary name of SM, are

extremely useful tools or instruments, long awaited for. The proof is in the large number and thediversity of periodical functions introduced in mathematics, and, sometimes, the complex way of

reaching them, by trying the substitution of the circle with other curves, most of them closed.

To obtain new special, periodical functions, it has been attempted the replacement of the

trigonometric circle with the square or the diamond. This was the proceeding of Prof. Dr. Math.

Valeriu Alaci, the former head of the Mathematics Department of Mechanics College from

Timioara, who discovered the square and diamond trigonometric functions. Hereafter, themathematics teacherEugenVisa introduced the pseudo-hyperbolic functions, and the mathematicsteacher M. O. Enculescu defined the polygonal functions, replacing the circle with an n-sides

polygon; for n = 4 he obtained the square Alaci trigonometric functions. Recently, the mathematician,Prof. Malvina Baica, (of Romanian origin) from the University of Wisconsin together with Prof.

Mircea Crdu, have completed the gap between the Euler circular functions and Alaci squarefunctions, with the so-called Periodic Transtrigonometric functios.

The Russian mathematician Marcusevici describes, in his work Remarcable sine functions

the generalized trigonometric functions and the trigonometric functions lemniscates.

Even since 1877, the German mathematician Dr. Biehringer, substituting the right trianglewith an oblique triangle, has defined the inclined trigonometric functions. The British scientist of

Romanian origin EngineerGeorge (Gogu) Constantinescu replaced the circle with the evolventand

defined the Romaniantrigonometric functions:Romanian cosine and Romanian sine, expressed by

Corand Sirfunctions,which helped him to resolve some non-linear differential equations of theSonicity Theory, which he created. And how little known are all these functions even in Romania!

Also the elliptical functions are defined on an ellipse. A rotated one, with its main axis along

Oy axis.How simple the complicated things can become, and as a matter of fact they are! This

paradox(ism) suggests that by a simple displacement/expulsion of a point from a center and by the

apparition of the notion of the ex-center, a new world appeared, the world of EMand, at the sametime, a new Universe, the SMUniverse.

Notions like Supermathematics Functions and Circular Ex-centric Functions

appeared on most search engines like Google, Yahoo, AltaVista etc., from the beginning of the

Internet. The new notions, like quadrilobequadrilobas, how the ex-centrics are named, and whichcontinuously fill the space between a square circumscribed to a circle and the circle itself were

included in the Mathematics Dictionary. The intersection of the quadriloba with the straight line dgenerates the new functions called cosinequadrilobe-ic andsine quadrilobe-ic.

The benefits ofSM in science and technology are too numerous to list them all here. But we

are pleased to remark that SM removes the boundaries between linear and non-linear; the linear

belongs to CM, and the non-linear is the appanage ofEM, as between ideal and real, or as betweenperfection and imperfection.

It is known that the Topology does not differentiate between a pretzel and a cup of tea.Well,

SM does not differentiate between a circle(e = 0) and a perfect square (s = 1), between a circle


14/133

13

and a perfect triangle, between an ellipse and a perfect rectangle, between a sphere and a perfectcube, etc. With the same parametric equations we can obtain, besides the ideal forms ofCM (circle,

ellipse, sphere etc.), also thereal ones (square, oblong, cube, etc.). Fors [-1,1], in the case of ex-centric functions of variable, as in the case of centric functions of variable , fors[-,+], it can be obtained an infinity of intermediate forms, for example, square, oblong or cube with rounded

corners and slightly curved sides or, respectively, faces. All of these facilitate the utilization of thenew SM functions for drawing and representing of some technical parts, with rounded or splayed

edges, in the CAD/ CAM-SM programs, which dont use the computer as drawing boards any more,

but create the technical object instantly, by using the parametric equations, that speed up theprocessing, because only the equations are memorized, not the vast number of pixels which define the

technical piece.

The numerous functions presented here, are introduced in mathematics for the first time,

therefore, for a better understanding, the author considered that it was necessary to have a short presentation of their equations, such that the readers, who wish to use them in their applications

development, be able to do it.

SM is not a finished work; its merely an introduction in this vast domain, a first step, the

authors small step, and a giant leap for mathematics.The elevated circular SM functions (ELC-SMF), named this way because by the

modification of the numerical ex-centricity s the points of the curves of elevated sine functions sel as of the elevated circular function elevated cosine celis elevating in other words it rises on thevertical, getting out from the space {-1, +1] of the other sine and cosine functions, centric or ex-

centric. The functions cexandsex plots are shownin Fig. 3, where it can be seen that the pointsof these graphs get modified on the horizontal direction, but all remaining in the space

[-1,+1], named the existence domain of these functions.

The functions cel and sel plots can be simply represented by the products:

cel 1,2 = rex1,2 . cos and Cel1,2= Rex 1,2. cos

sel 1,2 = rex 1,2 . sin and Sel 1,2 = Rex 1,2. sin

and are shown Fig. 4.

The exotic circular functionsare the most general SM, and are defined on the unity circle

which is not centered in the origin of the xOy axis system, neither in the eccenterS,but in a certain

point C (c,) from the plane of the unity circle, of polar coordinates (c, ) in the xOy coordinatesystem.

Many of the drawings from this album are done with EC-SMF of ex-center variable and with

arcs that are multiples of (n.). The used relations for each particular case are explicitly shown, inmost cases using the centric mathematical functions, with which, as we saw, we could express all SM

functions, especially when the image programs cannot use SMF. This doesnt mean that, in the future,the new math complements will not be implemented in computers, to facilitate their vast utilization.


15/133

14

Fig. 3,a The ex-centric circular

supermathematicsfunction

(EC-SMF) ex-centric cosine ofcex

for = 0, [0, 2]

Fig. 3,b The ex-centric circular

supermathematicsfunction

(EC-SMF) ecentric sine ofsex for

= 0, [0, 2]Numerical ex-centricity s = e/R [ -1, 1]

The computer specialists working in programming the computer assisted design software

CAD/CAM/CAE, are on their way to develop these new programs fundamentally different, becausethe technical objects are created with parametric circular or hyperbolic SMFs, as it has been

exemplified already with some achievements such as airplanes, buildings, etc. in

http://www.eng.upt.ro/~mselariuand how a washer can be represented as a toroid ex-centricity (or asan ex-centric torus), square or oblong in an axial section, and, respectively, a square plate with a

central square hole can be a square torus of square section. And all of these, because SMdoesnt

make distinction between a circle and a square or between an ellipse and a rectangle, as we mentionedbefore.

But the most important achievements in science can be obtained by solving some non-linearproblems, because SMreunites these two domains, so different in the past, in a single entity. Among

these differences we mention that the non-linear domain asks for ingenious approaches for each

problem. For example, in the domain of vibrations, static elastic characteristics (SEC) soft non-linear

(regressive) or hard non-linear (progressive) can be obtained simply by writing y = m. x, where m is


16/133

15

not anymorem = tan as in the linear case (s = 0 ), but m = tex1,2 and depending on the numericalex-centricity s sign, positive or negative, or for S placed on the negative x axis ( =) or on thepositive x axis ( = 0), we obtain the two nonlinear elastic characteristics, and obviously for s=0 wellobtain the linear SEC.

1 2 3 4 5 6

-1

-0.5

0.5

1

1.5

2

1 2 3 4 5 6

-2

-1.5

-1

-0.5

0.5

1

1 2 3 4 5 6

-1

-0.5

0.5

1

1.5

1 2 3 4 5 6

-1.5

-1

-0.5

0.5

1

Fig. 4,a ELC-SMF elevated cosine of - cel, fors [-1, +1], = 0, [0, 2].

Fig. 4,b ELC-SMFelevated sine of - sel, fors[-1, +1], = 1, [0, 2].

Due to the fact that the functions cex and sex , as well Cex and Sex and theircombinations, are solutions of some differential equations of second degree with variable coefficients,it has been stated that thelinear systems (Tchebychev) are obtained also fors = 1, and not onlyfors

= 0. In these equations, the mass ( the point M) rotates on the circle with a double angular speed =2. (reported to the linear system where s = 0 and = = constant) in a half of a period, and inthe other half of period stops in the point A(R,0) fore = sR= R or = 0and in A(R, 0) for e = s.R= 1, or = . Therefore, the oscillation period T of the three linear systems is the same andequal with T=/ 2. The nonlinear SEC systems are obtained for the others values, intermediates,ofs and e. The projection, on any direction, of the rotating motion of M on the circlewith radius R,

equal to the oscillation amplitude, of a variable angular speed = .dex ( afterdex function) isan non-linear oscillating motion.

The discovery of king function rex, with its properties, facilitated the apparition of ahybrid method (analytic-numerical), by which a simple relation was obtained, with only two terms,

to calculate the first degree elliptic complete integral K(k), with an unbelievable precision, with a

minimum of15 accurate decimals, after only 5 steps. Continuing with the next steps, can lead us to anew relation to compute K(k), with a considerable higher precision and with possibilities to expand

the method to other elliptic integrals, and not only to those. After 6 steps, the relation ofE (k) has the

same precision of computation.

s [0, 1]

s [ -1, 0]


17/133

16

The discovery of SMF facilitated the apparition of a new integration method, named

integration through the differential dividing.We will stop here, letting to the readers the pleasure to delight themselves by viewing the

drawings from this album.

[Translated from Romanian by Marian Niu and Florentin Smarandache]

Mircea Eugen elariu


18/133

17

Selariu SuperMathematics Functions

&

Other SuperMathematics Functions


19/133

18

D O U B L E C L E P S Y D R A

-0.5

0

0.5

-0.5

0

0.5

-1

-0.5

0

0.5

1

-0.5

0

0.5

M

=

=

=

sexz

ucexy

ucexx

sin.

cos.

, EccenterS(1,0) s = 1, = 0, t [0, 3]; u [0, 2]


20/133

19

S U P E R M A T H E M A T I C S F L O W E R S

0.5 1 1.5 2

0.5

1

1.5

2

0.5 1 1.5 2

0.5

1

1.5

2

X = dex (/`2), s = s0 cos 4 , = 0

Y = dex , s0 [0, 1], [ 0, 2 ]X = dex ( + /`2), s = s0 cos 6 , = 0

Y = dex , s0 [0, 1], [ 0, 2 ]

0.5 1 1.5 2

0.5

1

1.5

2

0.5 1 1.5 2

0.5

1

1.5

2

X = dex ( + /`2), s = s0 cos 8 , = 0

Y = dex , s0 [0, 1], [ 0, 2 ]

X = dex ( + /`2), s = s0 cos 15 , = 0

Y = dex , s0 [0, 1], [ 0, 2 ]


21/133

20

The Ballet of the Functions

0 1 2 3 4 5 6

-1

0

1

2

0 1 2 3 4 5 6

-0.4

-0.2

0

0.2

0.4

0.6

F(x) = sign(cos x).cos x / (1+ tann x)n, , n [0, 10] F (x) = sign(sin x).sin x / (1+ tann x)n, , n [0, 10]

0 1 2 3 4 5 6

-0.2

-0.1

0

0.1

0 1 2 3 4 5 6

-0.2

-0.1

0

0.1

0.2

F (x) = sign(cos x) tan x (1 + Abs(tan n x))n, n [0, 10] F (x) = sign(sin x) tan x (1 + Abs(tan n x))n, n [0, 10]

0 1 2 3 4 5 6

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0 1 2 3 4 5 6

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

F (x) = sign(sin x) tan x / (1 + Abs(tan x))n, n [0, 10] F (x) = sign(cos x) tan x / (1 + Abs(tan x))n, n [0, 10]


22/133

21

J a c u z z i

-2

-1

0

1

2-2

-1

0

1

2

-1

-0.5

0

0.5

1

-2

-1

0

1

2

-2

-1

0

1

2

-2

-1

0

1

2

-1

-0.5

0

0.5

1

-2

-1

0

1

2

M

==

=

==

]3,0[....1,.

]2,0[,..sin).1(

2sin,..cos).1(

ssexz

uucexy

usucexx

M

==

==

==

]3,0[.....................1,.

]2,0[,cos,.sin).1(

sin........,.........cos).1(

ssexz

uusucexy

usucexx

I m p l o d e d J a c u z z i-2

-1

0

1

2

-0.5

0

0.5

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

M

==

==

==

]3,0[.....................1,.

]2,0[,sin,.sin).1(

sin........,.........cos).1(

ssexz

uusucexy

usucexx


23/133

22

D a m a g e d p a r t o f T I T A N I C

-0.5

0

0.5

-2

-1

0

1

2

-1

-0.5

0

0.5

1

-0.5

0

0.5

-2

-1

0

1

2

M } ]2,0[],3,0[,1)cos.1(cos).1(

=

=

+=

=

us

sexz

ucexy

ucexx


24/133

23

K A Z A T C I O K (Russian Popular Dance)

-2 -1 1 2

-2

-1

1

2


25/133

24

F l y i n g B i r d 1

-2 -1 1

-2

-1

1

2

M

+===

==

3cos)8.0',0(.5sin1

3cos])1,0[,0,(.5sin12

2

sSsexy

scexx, S=s.cos5, ]2,0[ ,s ]1,0[

-1 -0.5 0.5 1 1.5

-1

-0.5

0.5

1

1.5

M ]2,0[],1,0[,5cos.)5cos(.)((sin1)8.0,(

)5cos(.)3(sin1),(2

2

=

+==

+=ssS

SsSssexy

SsSscexx


26/133

25

F l y i n g B i r d 2

-1 -0.5 0.5 1 1.5

-2

-1

1

2

M ]2,0[,0],1,0[,3cos9sin1)8.0,(

3cos.5cos.3sin1),(2

2

=

++==

+=s

ssexy

sscexx

-1 -0.5 0.5 1 1.5

-1.5

-1

-0.5

0.5

1

1.5

M ]2,0[,0],1,0[,5cos.3cos.9sin1)8.0,(

3cos.5cos.3sin1),(2

2

=

++==

+=s

sssexy

sscexx


27/133

26

M U L T I C O L O R E D S U N

-2 -1 1 2

-2

-1

1

2

M ]2,0[],1,1[,sin..20

cos..20

=

=

sdexy

dexx


28/133

27

R e d S u n

-2 -1 1 2

-2

-1

1

2

M ]2,0[],1,1[,sin..20

cos..20

=

=s

dexy

dexx


29/133

28

T h e d o u b l e N o z z l e f o r N A S A

-2-1

01

2

-2

-1

0

1

2

-2

0

2

-2

-1

0

1

2

-2-1

01

2

-1

0

1

-2

0

2

-1

0

1

-2-1

01

2

-2

-1

0

1

2

-2

0

2

-2

-1

0

1

2

--1

01

2

-2

-1

0

1

-2

0

2

-2

-1

0

1

M

=

=

=

sz

xny

xnx

3

sin..Re

cos..Re

, s [-1, 1], [0, 2], n = 2 ,3 ,4, 5.


30/133

29

T h e s u p e r m a t h e m a t i c s C o m e t

-4 -3 -2 -1 1

-2

-1

1

2

M ]2,0[),0],1,0[(,sin.

cos.

=

=

=sS

dexy

dexx


31/133

30

The Lake of Swans The Dance of Swords

0 1 2 3 4 5 6

-1

0

1

2

0 1 2 3 4 5 6

-1.5

-1

-0.5

0

0.5

1

1.5

nnx

xxsign

)tan1(

cos)(cos

+, n [0, 10]; x [0, 2]

x

xxsignntan1

cos)(

+, n [-10, 10]; x [0, 2]

The Nut Cracker The Decease of Swan

0 1 2 3 4 5 6

-1

-0.5

0

0.5

1

0 1 2 3 4 5 6

-0.4

-0.2

0

0.2

0.4

0.6

x

xxsignn 2tan1

5cos)(sin

+ , n [-10, 10]; x [0, 2] nnx

xxsign

)tan1(

sin)(sin

+ , n [0, 10]; x [0, 2]


32/133

31

T h e F l o w e r i n g

0.6 0.8 1.2 1.4

0.6

0.8

1.2

1.4

0.4 0.6 0.8 1.2 1.4 1.6

0.6

0.8

1.2

1.4

M

==

==

2,0[,,4cos.,

,4sin.,4cos.),

1

21

ssdexy

sssdexx

s [0, 1], = - /2, [0, 2,]

M

==

==

2,0[,,8cos.,

,8sin.,8cos.),

1

21

ssdexy

sssdexx

s [0, 1], = - /2, [0, 2,]

0.5 1 1.5 2

0.6

0.8

1.2

1.4

0.4 0.6 0.8 1.2 1.4 1.6

0.6

0.8

1.2

1.4

M

====

====

0,8cos.,2cos.,

2/,2sin.,2cos.,2/3

2

3

1

2

2

1

ssssdexy

ssssdexx

s [0, 1], [0, 2]

====

====

0,2cos.,2cos.,

2/,2sin.,2cos.,2/3

2

3

1

2

2

1

ssssdexy

ssssdexx

s [0, 1], [0, 2]


33/133

32

The supermathematics ring surface

-4

-2

0

2

4-4

-2

0

2

4

-1-0.5

00.5

1

-4

-2

0

2

4

-4

-2

0

2

4-4

-2

0

2

4

-1

-0.5

0

0.5

1

-4

-2

0

2

4

M

+=

+=

q

quqy

quqx

sin

sin).cos3(

cos).cos3(

s =1, = 0, [0, 2], u [, ]

-2

0

2

-2

0

2

-1

-0.5

0

0.5

1

-2

0

2

-2

0

2-2

0

2

-1

-0.5

0

0.5

1

-2

0

2

M

+=

+=

sex

ucexy

ucexx

sin).2(

cos).2(

s =1, = 0, [0, 2], u [0, 2]


34/133

33

The ex-centric sphere

-2

-1

0

1

2

-1

0

1-1

-0.5

0

0.5

1

-2

-1

0

1

2

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

M

=

=

=

sin.

sin.cos.

cos.cos.

dexz

udexy

udexx

S(s [0,1], =0) M

=

=

=

sin.cos

sin.cos.sin

cos.cos.cos

qz

uqy

uqx

S(s [0,1], =0)

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

M

=

==

uz

usexy

ucexx

sin

sin.

cos.

S(s [0,1], =0) M

=

==

uqz

uqy

uqx

sin.cos

sin.sin

cos.cos

S(s [0,1], =0)

[0, 2], u [, ], S[ s [ 0, 1], = 0]


35/133

34

T h e s u p e r m a t h e m a t i c s Screw Surface

-20

-10

0

10

20

-20

-10

0

10

-20

0

20

40

60

-20

-10

0

10

M

++===

++=

++=

)13

2sin(

13

8.4)0,1,

4(.8

)]13

2cos(5sin[

13

2

)]13

2cos(5cos[

13

2

ussexz

uy

ux

, u [0, 2], [0, 26]


36/133

35

T h e T r o j a n H o r s e

0

2

4

6

-1

-0.5

0

0.5

1

-4

-2

0

2

4

-1

-0.5

0

0.5

1

-4

-2

0

2

4

0

2

4

6

-1-0.5

00.5

1

-4

-2

0

2

4

-1-0.5

00.5

1

-4

-2

0

2

4

X = , y = s, Z = cosq [, S(s, )],

S[s

[ - 1, 1], = 0],

[0, 2]

X = , y = s, Z = sinq [, S(s, )],

S[s

[ - 1, 1], = 0],

[0, 2]


37/133

36

T h e A m p h o r a s

-2

-1

0

12

-1

0

1

-10

-7.5

-5

-2.5

0

-1

0

1

-1

0

1

-1

0

1

0

2

4

6

-1

0

1

M

=

=+=

=+=

]2,0[],5.0,6.3[,...9.0

sin.)2

cos,(.21

cos.)98.0,(.21

2

2

z

ssexssy

scexssx

M ]2,0[,

]2.2,0[

sin.Re

cos.Re

=

=

=

sz

xy

xx


38/133

37

B U D D H A

-4

-2

0

2

4-2

-1

0

1

2

-10

-7.5

-5

-2.5

0

-2

-1

0

1

2

M

=

=+=

=+=

]2,0[],5.0,6.3[,9.0

sin.)cos,(..214.1

cos.)98.0,(.212

2

2

1

2

ssz

sscexssy

sssexssx


39/133

38

J E T P L A N E

0

2

4

6

8-2

0

2

0

1

2

0

2

4

6

8


40/133

39

T R O U B L E D L A N Do r

D O U B L E A N A L Y N I C A L E X C E N T R I C F U N C T I O N

0

5

101

2

3

4

0

0.5

1

0

5

10

cex2a(x,S(s, = 0), ) = cos{ )]2

arcsin(2

[2

xxx }


41/133

40

S E L F - P I E R C E B O D Y

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

-1-0.5 0 0.5 1

-1

-0.5

0

0.5

1-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

-1 -0.5 0

0.5 1

-1

-0.5

0

0.5

1-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

-1

-0.5

0

0.51

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

-1

-0.5

0

0.51

-1

-0.5

0

0.5

1

M

=

=

==

])2,0[(.2sin

])2,0[(.2sin

)9.0,3(

scexz

ssexy

sbexx


42/133

41

H I L L S a n d V A L L E Y S

- 2

0

2

- 2

0

2

- 8

- 6

- 4

- 2

0

- 2

0

2

z = 10/(1 + 1.3 x2

+ y2)[sex[(x

2- y

2), S(s = 0.8, = 0)] Rex[(x2 y2), S(s = 0.64, = 0)],

x, y [ - , + ]

-2

0

2

0

1

2

3

-6

-4

-2

0

-2

0

2

z = 10/(1 + 1.3 x2 + y2)[sex[(x2 - y2 - 8), S(s = 0.8, = 0)] Rex[(x2 y2 - 8), S(s = 0.64, = 0)],x [ - , + ], y [ 0, + ],


43/133

42

B e r n o u l l i s L e m n i s c a t e, C a s s i n i s O v a l s a n d o t h e r s

0.4 0.6 0.8 1.2 1.4 1.6

0.5

1

1.5

2

M ]2,0[),2

],1,0[(2cos.,2(

)cos,(0

0

0

=

==

==sS

ssdexy

ssdexx


44/133

43

Continuous transforming of a circle into a haystack

0.4 0.6 0.8 1.2 1.4 1.6

0.2

0.4

0.6

0.8

1

1.2 1.4 1.6 1.8 2

0.4

0.6

0.8

1.2

1.4

1.6

M ]2,0[],1,0[

))0,cos(,(

))2

,cos)(,(0

0

0

===

===s

ssSdexy

ssSdexx


45/133

44

C y c l i c a l S y m m e t r y

0.4 0.6 0.8 1.2 1.4 1.6

0.25

0.5

0.75

1

1.25

1.5

1.75

0 . 4 0 . 6 0 . 8 1 . 2 1 . 4 1 . 6

0 . 2 5

0. 5

0 . 7 5

1

1 . 2 5

1. 5

1 . 7 5

0.5 1 1.5 2

0.5

1

1.5

2

M M ]2,0[],1,0[

))0,4cos(,(

))2

,4cos)(,(0

0

0

===

===s

ssSdexy

ssSdexx


46/133

45

S m a r a n d a c h e S t e p p e d F u n c t i o n s

2 4 6 8 10 12

2.5

5

7.5

10

12.5

15

6 8 10 12

-10

-5

5

10

F(n, s, ) = Ssf () = [ bex(, s = 1) dex ].s.dex n, n = 10, s = [0.2, 0.4, 0.6, 0.9, 1]


47/133

46

S C R I B B L I N G S W I T H . . . H E A D A N D T A L E

-4 -2 2 4

-4

-2

2

4

-4 -2 2 4

-4

-2

2

4

M ]2,0[],1,0[,32)1(

32)1(

=

+=s

sexsexy

cexcexx


48/133

47

Q U A D R I P O D

-2

0

2

-2

0

2

-1

-0.5

0

0.5

1

-2

0

2

z = cexq [x.y, S(s = 0.8, = 0)], x, y [, + ]


49/133

48

E X C E N T R I C S Y M M E T R Y

-4 -2 2 4

-4

-2

2

4

-4 -2 2 4

-4

-2

2

4

M ]2,0[),0],1,0[]0,1[()53sin(2.3

72.3

=+

=

+=andssS

bexsexy

cexcexx

-4 -2 2 4

-4

-2

2

4

M ]2,0[),0],1,1[()53sin(2.3

72.3

=+

=

+=sS

bexsexy

cexcexx


50/133

49

D R A C U L A S C A S T L E

-2

0

2

-2

0

2

-1

-0.5

0

0.5

1

-2

0

2

-1

0

1

-2

0

2

0

2.5

5

7.5

10

-1

0

1

Z = cosq (x.y), s = 0.8,

x, y [, + ]Z = 1 / cosq (x.y). cos (x.y), s = 0.8,

x, y [, + ]

-2

0

2

-2

0

2

-4

-2

0

2

4

-2

0

2

Z = 1 / cosq (x.y), s = 0.8, x,y [, + ]


51/133

50

T U N I N G F O R K

-4

-2

0

2

4

-10

0

10

-1-0.5

00.5

1

-4

-2

0

2

4


52/133

51

d e x - O I D S

01

2

3

-1-0.5

0

0.51

0

0.5

1

1.5

2

01

2

3

0

1 2

3

-

-0.5

00.5

1

0

0.5

1

1.5

2

0

1 2

3

z = dex(x, y), x [0, ], S( s y [-1,1], = 0)

r e x O I D S

0

1

23

- 1

- 0 . 5

00 . 5

1

0

0 . 5

1

1 . 5

2

0

1

23

z = Rex(x, y), x [0, ], S( s y [-1,1], = 0)


53/133

52

E x c e n t r i c g e o m e t r y , p r i s m a t i c s o l i d s

0

0.5

1

1.5

2

0

0.5

1

1.5

2

0

0.5

1

1.5

0

0.5

1

1.5

2

0

0.5

1

1.5

2

0

0.5

1

1.5

2

0

0.5

1

1.5

-1.5

-1

-0.5

0

0

0.5

1

1.5

2

0

0.5

1

1.5

M ]0,1[..]..1,0[],2,0[,

5.1

)2/,4(

)0,4(

=

==

==

sors

sz

dexy

dexx

-1-0.5

0

0.51

-1

-0.5

0

0.5

1

-3

-2

-1

0

1

-1

-0.5

0

0.5

1

-1-0.5

00.5

1

-1

-0.5

0

0.5

1

-3

-2

-1

0

1

-1

-0.5

0

0.5

1

M1

=

=

=

2

sin

cos.

1

1

1

sz

qy

qsx

, M2

=

=

=

sz

qy

qx

sin

cos

2

2

, [0, 2], S( = 0, s [-1,1])


54/133

53

V a s e

-1

01

-1

0

1

-10

-7.5

-5

-2.5

0

-1

0

1

-2-1

01

2

-1

0

1

-10

-7.5

-5

-2.5

0

-1

0

1

M ]2,0[,

]0,6.3[,

sin).(Recos).(Re

=

==

sz

sxysxx

M

=

=+==

=+==

]2,0[],2/,6.3[,.9.0

)5cos,(.21sin),(Resin

]98.0,[.21cos),(Recos

222

1

2

1

ssz

sssexsssxy

sscexsssxx


55/133

54

H E X A G O N A L T O R U S

-4

-2

0

2

4-4

-2

0

2

4

-1

-0.5

0

0.5

1

-4

-2

0

2

4

-4

-2

0

2

4-4

-2

0

2

4

-1

-0.5

0

0.5

1

-4

-2

0

2

4

M ]2,0[],2,0[,

]0,1(,[

sin]).0,1(,[3(

cos]).0,1(,[3(

111

111

111

===

==+=

==+=

s

sSssexz

sSscexy

sSscexx


56/133

55

O p e n s q u a r e t o r u s M ]2.2,0[],2,0[,)sin)cos3(

cos).cos3(

=

+=

+=

s

sz

qsy

qsx

-4

-2

0

2

4

-4

-2

0

2

4

0

2

4

6

-4

-2

0

2

4

-4

-2

0

2

4

S q u a r e t o r u s M ]2.2,0[],2,0[,

sin

)1,(sin)cos3(

)1,(cos).cos3(

1

1

=

=+=

=+=

s

sz

sqsy

sqsx

-4

-2

0

2

4-4

-2

0

2

4

-1

-0.5

0

0.5

1

-4

-2

0

2

4


57/133

56

D o u b l e s q u a r e t o r u s ]2,0[,,

)1,(sin

sin).cos1/)1,(cos3(

cos).sin1/)1,(cos3(2

2

=

+=

+=

s

qz

sqy

sqx

-4

-2

0

2

4-4

-2

0

2

4

-1

-0.5

0

0.5

1

-4

-2

0

2

4

-2

0

2

-2

0

2

-1

-0.5

0

0.5

1

-2

0

2

S q u a r e t o r u s ]2,0[,

)1,(sin

sin)].1,(cos3[

cos)].1,(cos2[

=

+=

+=

s

sqz

sqy

sqx


58/133

57

S i n u o u s C o r r u g a t e W a s h e r s

o r

N a n o - P e r i s t a l t i c E n g i n e

-4

-2

0

2

4

-4

-2

0

2

4

0

1

2

3

-4

-2

0

2

4

M

+

++=

+=

+=

30)

3/208sin(2.0)1,(sin3.0

]sin).1,(cos3[

]cos)].1,(cos3[

sqz

sqy

sqx


59/133

58

I G L O O M]2,0[,,

1.0

),(sin4

cos),(cos4

sin.22

),(sin4

sin),(cos4

cos.22

4

=

=

=

s

sz

sqsqsy

sqsqsx

-1

0

1

-1

0

1

0

0. 5

1

1. 5

-1

0

1

T h e m a g i c c a r p e t ]1,1[],2,0[)),0,(,( 2 === syxysSxbexz

0

2

4

6 -1

- 0 . 5

0

0. 5

1

- 0 . 5

0

0. 5

0

2

4

6

- 0 . 5

0

0. 5


60/133

59

Multiple Ex Centric Circular SuperMathematics Functions

1 2 3 4 5 6

-1

- 0 . 5

0. 5

1

sex with dauble ex centre: y = s2ex =sin[-arcsin[s.sin]-arcsin[s.sin[-arcsin[s.sin ]]]]

ex centre S( s [-1, 1], = 0 ), [0, 2]

1 2 3 4 5 6

-1

-0.5

0. 5

1

y = sin[ arctan[s.sin2 /Rex 2 ]]]

1 2 3 4 5 6

- 1

- 0 . 5

0 . 5

1

y = cos[ arctan(s.sin2)] / 2cos.sa


61/133

60

E X C E N T R I C T O R U S R I N G

M

===+=

=+=

],0[]2,0[],2,0[,

)8.0,(

sin)]}sin.9.0,2(cos[2{

cos)]}8.0,2(cos[2{

u

ssexz

uusbexy

usbexx

-2

0

2

-2

0

2

-1-0.5

0

0.5

1

-2

0

2

-2

0

20

1

2

3-1

-0.5

0

0.5

1

-2

0

2

-1

-0.5

0

0.5

1

M

==

=+=

=+=

],0[

]2,0[],2,0[,

)8.0,(

sin)]}sin.9.0,3(cos[2{

cos)]}8.0,3(cos[2{

u

ssexz

uusbexy

usbexx

-2

0

2

-2

0

2

-1

-0.5

0

0.5

1

-2

0

2

-3

-2

-1

0

-2

0

2

-1

-0.5

0

0.5

1

-3

-2

-1

0

-2

0

2


62/133

61

H Y P E R S O N I C J E T A I R P L A N E

M ]2,0[],0],1,0[(,

Re

)2/sin(.1(Re

sin.1

=

+=

=

=

sS

x

sz

xsy

sx

-10

-7.5

-5

-2.5

00

1

2

30

1

2

3

-10

-7.5

-5

-2.5

0

0

1

2

3

0

2

4

6

8

-0.5

0

0.5

-0.5

0

0.5

0

2

4

6

8

-0.5

0

0.5

S ( s [ 0, 0.8], = 0)


63/133

62

P L U M P V A S E

M ]2,0[),0,3.2,0[(,.sin..cos.25.Re

cos.cos.25.Re2

2

=

=

+=

+=

sS

sz

ssxsy

ssxsx

-2-1

01

2

-2

0

2

0

2

4

6

-2

0

2

-2

-1

0

1

2

-2

0

2

0

2

4

-2

-1

0

1

2

-2

0

2

- 2

0

2

- 2

0

2

0

2

4

- 2

0

2

- 2

0

2


64/133

63

A R R O W S

-0.4-0.2

00.2

0.4

-1

0

1

2

-0.5

-0.25

0

0.25

0.5

-0.4-0.2

00.2

0.4

-1

0

1

2

-0.4-0.20

0.20.4

-1

0

1

2

-0.5

-0.25

0

0.25

0.5

-0.4-0.20

0.20.4

-1

0

1

2

-0.5

-0.25

0

0.25

0.5

-0.4-0.2

0

0.2

0.4

-1

0

1

2

-0.5

-0.25

0

0.25

0.5

-0.4-0.2

0

0.2

0.4

-1

0

1

2

-0.4

-0.2

0

0.20.4

-1

0

1

2

-0.5

-0.25

0

0.25

0.5

-0.4

-0.2

0

0.20.4

-1

0

1

2

M ]2,0[),0),2

,2

[(,

sin.5.0

cos)1,0,().1,0,(2

cos.cos1).1,0,( 2

=

=

=

=

sS

sz

sdelcexy

ssexx


65/133

64

H Y P E R B O L I C Q U A D R A T I C C Y L I N D E R 1

R I G H T : n = 4, m = 1 R O T A T E D n = 1, m = 0.5

00.5

11.5

2

0

0.5

1

1.5

2

-1

0

1

0

0.5

1

1.5

2

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

-1

0

1

-1

-0.5

0

0.5

1

M ]2,0[),0],1,1[(,

5.1

)0,,(sin

)0,,(cos

=

=

====

sS

sz

snqy

snqxm

m


66/133

65

H Y P E R B O L I C Q U A D R A T I C C Y L I N D E R 2

0

1

2

0

0 . 5

1

1 . 5

2

- 1

0

1

0

0 . 5

1

1 . 5

2

0

0 . 5

1

1 . 5

2

0

1

2

- 1

0

1

0

0 . 5

1

1 . 5

2

0

1

2


67/133

66

E X - C E N T R I C F U L L S P R I N G

-1

0

1

-1

0

1

-1

0

1

2

3

-1

0

1

M

===

==

===

==

))0,1(,4

(

cos3.0

)cos3.0

))0,1(,4

(2.0

)0,1(,4

(2.0

sSaexz

y

x

sSaex

sSaex


68/133

67

E X - C E N T R I C E M P T Y S P R I N G

-20

-10

0

10

-10

0

10

0

20

40

-10

0

10


69/133

68

E X C E N T R I C P E N T A G O N H E L I X

-50

0

50

-50

0

50

0

50

100

150

-50

0

50


70/133

69

C Y L I N D E R S w i t h C O L L A R S

-20

2

-2

0

2

-5

0

5

-2

0

2

-2

0

2

-2

0

2

-5

0

5

-2

0

2


71/133

70

U n i c u r s a l S u p e r m a t h e m a t i c s F u n c t i o n s 1

- 2 2 4 6 8

- 1

- 0 . 5

0 . 5

1

]2

3,2

[,

8.4

8.3

5.3

,

4

5.2

1

,,

)cos.2cos

sin.2cos.arctansin(

cos.2cos

sin.2cos.arctan

==

++=

++=

cba

cb

ay

cb

ax

O l d w o m a n f r o m C a r p a t i M o u n t a i n ( R o m a ni a )

2 4 6 8 10 12

-1

-0.5

0.5

1


72/133

71

U n i c u r s a l S u p e r m a t h e m a t i c s F u n c t i o n s 2

2 4 6 8 1 0 1 2

-1

- 0 . 5

0. 5

1

2 4 6 8 10 12

-1

-0.5

0.5

1


73/133

72

U n i c u r s a l Su p e r m a t h e m a t i c s F u n c t i o n s 3

1 2 3 4 5 6

- 1

- 0 . 5

0 . 5

1

W a l k i n g P i n g u i n s

-2.5 2.5 5 7.5 10 12.5

-1

-0.5

0.5

1


74/133

73

T o D o u b l e a n d S i m p l e C a n o e

- 0 . 5

- 0 . 2 5

0

0 . 2 5

0 . 5

- 1

0

1

2

- 0 . 5

- 0 . 2 5

0

0 . 2 5

0 . 5

- 0 . 5

- 0 . 2 5

0

0 . 2 5

0 . 5

- 1

0

1

2

- 1

- 0 . 5

0

0 . 5

1

- 1

0

1

2

- 0 . 5

- 0 . 2 5

0

0 . 2 5

0 . 5

- 1

- 0 . 5

0

0 . 5

1


75/133

74

R o m a n i a n f o l k d a n c e

-4

-2

0

2

4

-4

-2

0

2

4

-5

-2.5

0

2.5

5

-4

-2

0

2

4


76/133

75

P i l l o w

2

2 . 5

3

3 . 5

4

2

2 . 5

3

3 . 5

4

- 1

- 0 . 5

0

0 . 5

1

2

2 . 5

3

3 . 5

4

2

2 . 5

3

3 . 5

4

2

2.5

3

3.5

4

2

2.5

3

3.5

4

-1

-0.5

0

0.5

1

2

2.5

3

3.5

4

2

2.5

3

3.5

4


77/133

76

T e r r a w i t h M a r k e d M e r i d i a n s

2

2.5

3

3.5

4

2

2.5

3

3.5

4

-1

-0.5

0

0.5

1

2

2.5

3

3.5

4

2

2.5

3

3.5

4

M

=

=

+=

sexz

ucexy

cexx

sin.

cos.3

, S( s u [0, 3 /2], [0, 2]


78/133

77

S u p e r m a t h e m a t i c s C o l u m n s

??

x = cosq t

y = sinq t , t [0, 2]z u [-1,1]

-1

-0.50

0.51

-1

-0.5

0

0.5

1

-2

0

2

4

-1

-0.5

0

0.5

1

- 0 . 5- 0 . 2 5

00 . 2 5

0 . 5

- 0 . 5

- 0 . 2 5

0

0 . 2 5

0 . 5

0

2

4


79/133

78

A e r o d y n a m i c S o l i d

-

0

1

- 1

- 0 . 5

0

0 . 5

1- 1

- 0 . 5

0

0 . 5

1

- 1

- 0 . 5

0

0 . 5

1

- 1

0

1

- 1

- 0 . 5

0

0 . 5

1

- 1

- 0 . 5

0

0 .5

1

- 1

- 0 . 5

0

0 .5

1


80/133

79

H a l v i n g C u r v e

x = cosq , y = sinq , with numerical ex-center : S( s, = 0 ), [0, 8 ]S (s = 1, = 0) S (s = 0.89, = 0)

5 10 15 20 25

-1

-0.5

0.5

1

5 10 15 20 25

-1

-0.5

0.5

1

S (s = 0, = 0) S (s = 0.5, = 0)

5 10 15 20 25

-1

-0.5

0.5

1

5 10 15 20 25

-1

-0.5

0.5

1

S (s = 0.5, = 0) , x x2 S (s = 0.5, = 0), y y 2

5 10 15 20 25

-1

-0.5

0.5

1

5 10 15 20 25

-1

-0.5

0.5

1


81/133

80

The crook lines (s 0) - a generalization of straight lines ( s = 0 )

y = m. aex (x, S )=m{x-arcsin [s. sin(x - )]}, Numerical ex centre S (s, ) fixed

m = 1, S (s [- 1, 0], = 0 ) m = 1, S (s [0, +1], = 0 )

-3 -2 -1 1 2 3

-3

-2

-1

1

2

3

-3 -2 -1 1 2 3

-3

-2

-1

1

2

3

y = m. aex (x, S), Numerical ex centre S (s, ) variable: s =xx

s

sin.cos

0

- 3 - 2 - 1 1 2 3

- 4

- 2

2

4

S ( s = s0 /cos x .sin x s0 [- 1, 0], = 0 )


82/133

81

The crook lines (s 0) - a generalization of straight lines ( s = 0 )

y = m. aex (x, S )=m{x-arcsin [s. sin(x - )]}, Numerical ex centre S (s, ) variable

m = 1, S (s = s0.cos2x, s0 [- 1, 0], = 0 ) m = 1, S (s= s0.cos2x, s0 [0, +1], = 0 )

-3 -2 -1 1 2 3

-3

-2

-1

1

2

3

-3 -2 -1 1 2 3

-3

-2

-1

1

2

3

m = 1, S (s= s0.cos2x, s0 [-1, +1], = 0 ), x [ - 3 / 2, 3 / 2 ]

- 4 - 2 2 4

- 6

- 4

- 2

2

4

6


83/133

82

The crook lines (s 0, s = 1) - a generalization of straight lines ( s = 0 )

y = m. aex (x, S )= m{x-arcsin [s. sin(x - )]}, Numerical ex centre S (s, ) fixed

-3 -2 -1 1 2 3

-4

-2

2

4

Numerical ex centre S (s, ) fixed : S (s [ - 1 , 1 ] , = 1 )


84/133

83

A R A B E S Q U E S 1

- 1 - 0 . 5 0 . 5 1

- 1

- 0 . 5

0 . 5

1

- 1 - 0 . 5 0 . 5 1

- 1

- 0 . 5

0 . 5

1


85/133

84

S T A R S

x = Dex 10 . Rex 10 .cos y = Dex 10 . Rex 10 .sin

S( s [-1,1 ] , = 0), [0, 2]

-1 -0.5 0.5 1

-1.5

-1

-0.5

0.5

1

1.5

-1 -0.5 0.5 1

-1.5

-1

-0.5

0.5

1

1.5

- 2 - 1 . 5 - 1 - 0 . 5 0 . 5 1 1 . 5

- 1 . 5

- 1

- 0 . 5

0 . 5

1

1 . 5

x = (1-cos 5 ) cos / Rex 10 .y = (1- cos 5 ) sin /Rex 10 .


86/133

85

H y s t e r e t i c C u r v e s 1

-4 -2 2 4

-1

-0.5

0.5

1

x = 1

M

}00.1,95,0,9.0,8.0,6.0,4.0,2.0{

]2

,2

[),0,5.0

1],1,0[(

,

cos1

)sin())1,(,(sin

sin1

)cos())1,(,(cos

22

22

=

=

=

===

===

s

sS

ssSqy

ssSqx

yx

y

x

-4 -2 2 4

-1

-0.5

0.5

1

x =0.5


87/133

86


-4 -2 2 4

-4

-2

2

4

x = 0.5; y=0.5=

-4 -2 2 4

-4

-2

2

4

x = 0.5; y= 1


88/133

87


-4 -2 2 4

-4

-2

2

4

x = 0.5; y= 0.5

-4 -2 2 4

-4

-2

2

4

x = 0.2; y=0.5


89/133

88

E x C e n t r i c C i r c u l a r C u r v e s w i t h E x C e n t r i c V a r i a b l e

- 1 - 0 . 5 0 . 5 1

- 1

- 0 . 5

0 . 5

1

0.1 S t e p

- 1 - 0 . 5 0 . 5 1

- 1

- 0 . 5

0 . 5

1

0 . 2 S t e p


90/133

89

F i l i g r e e 1

- 1 - 0 . 5 0 . 5 1

- 1

- 0 . 5

0 . 5

1

s 0.1 u [-1, 1] with 0.1 S t e p

- 1 - 0 . 5 0 . 5 1

- 1

- 0 . 5

0 . 5

1

s 0.1 u [-1, 1] with 0.2 S t e p


91/133

90

F I L I G R E E 2

- 1 - 0 . 5 0 . 5 1

- 1

- 0 . 5

0 . 5

1

s 0.1 u [0, 1] with 0.1 S t e p s 0.1 u [-1, 0] with 0.1 S t e p

-1 -0.5 0.5 1

-1

-0.5

0.5

1

-1 -0.5 0.5 1

-1

-0.5

0.5

1


92/133

91

U s h a p e d C u r v e s i n 2 D a n d 3 D

y = arctan{1/[sex . Abs[cex ]]}, S(s [-1,1] , = 0), [ 0, 2]

1 2 3 4 5 6

-60

-40

-20

20

40

60

0

2

4

6

-0.5

0

0.5

-1

0

1

-1

0

1


93/133

92

E x p l o s i o n s

-1 -0.5 0.5 1

-1

-0.5

0.5

1

-1 -0.5 0.5 1

-1

-0.5

0.5

1

]2,0[

)0],1,0[(,

10Re

sin.8cos.Re

cos.10cos.

=

=

= sS

x

sy

x

sx

]2,0[

)0],1,0[(,

8Re

sin.7cos.Re

cos.10cos.

=

=

= sS

x

sy

x

sx


94/133

93

S p a t i a l F i g u r e

M ]2,0[),0],1,0[(,

)]cos3(,2[3

5Re

5

2

0

=

==

=

=

sS

ssSCexz

yx

bexx

-1

0

1

-2

0

2

-2

0

2

-1

0

1

-2

0

2


95/133

94

P l a n e t s a n d s t a r s

-1 -0.5 0.5 1

-1

-0.5

0.5

1

-2 -1 1 2

-2

-1

1

2

-2 -1 1 2

-2

-1

1

2


96/133

95

E x C e n t r i c C i r c l e ( n = 1 ) a n d A s t e r o i d ( n = 2, 4 , 6 )

M ]2,0[),0],1,1[(,

=

=

=sS

sexy

cexxn

n

-1 -0.5 0.5 1

-1

-0.5

0.5

1

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

n = 1 n = 2

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

n = 4 n = 6


97/133

96

E x C e n t r i c A s t e r o i d ( n = 3, 5 ,7 ,9)

]2,0[),0],1,1[(,

=

=

=sS

sexy

cexxn

n

-1 -0.5 0.5 1

-1

-0.5

0.5

1

-1 -0.5 0.5

-1

-0.5

0.5

1

n = 3 n = 5

-0.4 -0.2 0.2 0.4

-0.4

-0.2

0.2

0.4

-0.2 -0.1 0.1 0.2

-0.2

-0.1

0.1

0.2

n = 7 n = 9


98/133

97

E x C e n t r i c L e m n i s c a t e s

M { x = dex[, S(s = s0. cos, = - / 2]; y = dex[, S( s = s0.cos 2, = 0)] }, s0 [0, 1], [0, 2]

0.4 0.6 0.8 1.2 1.4 1.6

0.5

1

1.5

2


99/133

98

B u t t e r f l y w i t h S y m m e t r i c a l C e n t e r 1

M { x = dex[, S(s = s0. cos, = - / 2]; y = dex[, S( s = s0.sin 2, = 0)] }, s0 [0, 1], [0, 2]

0.4 0.6 0.8 1.2 1.4 1.6

0.5

1

1.5

2


100/133

99


M { x = dex[, S(s = s0. cos, = - / 2]; y = dex[, S( s = s0.sin 2, = 0)] }, s0 [0, 1], [0, 2]

0 . 6 0 . 8 1 . 2 1 . 4

0 . 5

1

1 . 5

2


101/133

100


0 . 6 0 . 8 1 . 2 1 . 4

0 . 2 5

0 . 5

0 . 7 5

1

1 . 2 5

1 . 5

1 . 7 5


102/133

101

B u t t e r f l y R a p i d l y F l a p p i n g t h e W i n g s

0.5 1 1.5 2

0.5

1

1.5

2


103/133

102

F l o w e r w i t h F o u r P e t a l e s

0.25 0.5 0.75 1 1.25 1.5 1.75

0.25

0.5

0.75

1

1.25

1.5

1.75


104/133

103

E c C e n t r i c P y r a m i d

0.5

1

1.5

0

0.5

1

1.5

2

-1

-0.5

0

0.5

1

0.5

1

1.5

0

0.5

1

1.5

2


105/133

104

A e r o d y n a m i c P r o f i l e w i t h S u p e r m a t h e m a t i c s F u n c t i o n s 1

M { x = 2 cex( + / 6, S( s = 0.6, = 0)), y = 2 sex(( + / 6, S( s [ -1, 0], = 0) )}, [0, 2]

-2 -1 1 2

-1

-0.5

0.5

1

M { x = 2 cex( + / 6, S( s = 0.6, = 0)), y = sex(( + / 6, S( s [ 0, 1], = 0) )}, [0, 2]

-2 -1 1 2

-1

-0.5

0.5

1

M { x = 2 cex( + / 6, S( s = 0.2, = 0)), y = sex(( + / 2 , S( s [ 0, 0.9], = 0) )}, [0, 2]

- 1 - 0 . 5 0 . 5 1

- 1

- 0 . 5

0. 5

1


106/133

105

A e r o d y n a m i c P r o f i l e w i t h S u p e r m a t h e m a t i c s F u n c t i o n s 2

-1 -0.5 0.5 1

-1

-0.5

0.5

1

M { x = 2 cex( + / 6, S( s = 0.2, = 0)), y = sex(( + / , S( s [ 0, 0.9], = 0) )}, [0, 2]- 2

- 1 . 5

- 1

- 0 . 5

0- 1

- 0 . 5

0

- 2

- 1 . 5

- 1

- 0 . 5

0

- 1

- 0 . 5

0


107/133

106

S A L T C e l l a r

-2

-1

0

1

2

-2

-1

0

1

-2

-1

0

1

2

-2

-1

0

1

2

-2

-1

0

1

M ]2,0[),0],1,1[(,

.2

)2/(

1,2

2,1

=+

=

=

=

sS

sz

rexy

rexx


108/133

107

S U P E R M A T H E M A T I C S T O W E R

-5

-2.5

0

2.5

5

-5

0

5

-4

-2

0

2

4

-5

-2.5

0

2.5

5

-5

0

5

M ]2,0[),0],2,0[(,

sin..2

)).0,1(,().cos3(

))0,1(,().3(

=

==+=

==+=

sS

sdexs

dexsSsexsy

sScexdexsx


109/133

108

THE CONTINUOUS TRANSFORMATION OF A RIGHT TRIANGLE INTO ITS HYPOTENUSE

-1 -0.5 0.5 1

-1

-0.5

0.5

1

-1 -0.5 0.5 1

-1

-0.5

0.5

1

x=cex(, S(s, = 0)), y=cex(, S(-s, =0)), S(s [0,1], = 0), [ 0, ]

THE CONTINUOUS TRANSFORMATION OF QUADRATE (Rx = Ry)

OR RECTANGLE (Rx Ry) INTO ITS DIAGONAL

-1 -0.5 0.5 1

-1

-0.5

0.5

1

x = Rx cex(, S(s, = 0)), y = Ry cex(, S(-s, =0)), S(s [ -1, 1 ], = 0), [ 0, ]


110/133

109

Mo d i f i e d c e x a n d s e x

]2,0[),0],1,1[(

),2Re

2sin.arctansin(

22

=

+==

sS

x

sMsexy

]2,0[),0],1,1[(

),2Re

2sin.arctancos(

22

=

+==

sS

x

sMcexy

1 2 3 4 5 6

-1

-0.5

0.5

1

1 2 3 4 5 6

-1

-0.5

0.5

1

]2,0[),0],1,1[(

),2Re

2sin.arctancos(

21

=

==

sS

x

sMcexy

]2,0[),0],1,1[(

),2Re

2sin.arctansin(

21

=

==

sS

x

sMsexy

1 2 3 4 5 6

-1

-0.5

0.5

1

1 2 3 4 5 6

-1

-0.5

0.5

1

1 2 3 4 5 6

-1

-0.5

0.5

1

1 2 3 4 5 6

-1

-0.5

0.5

1


111/133

110

S i n u o u s S u r f a c e w i t h A n a l i t i c a l S u p e r m a t h e m a t i c s F u n c t i o n s

]4,20[],2.1,4[],3

sin5

cos21.0arcsin[),( =

x

xxxf

-4

-2

0

25

10

15

20

-0.2

-0.1

0

0.1

0.2

-4

-2

0

2

S e c t i o n f o r = 8 and x [ - 5, 10 ]

- 1 0 1 0 2 0 3 0

- 0 . 2

- 0 . 1

0 . 1

0 . 2


112/133

111

S u p e r m a t h e m a t i c s S p i r a l

x = 0.3 cos Exp[0.2 (0.25 arcsin(sin0.25 )y = 0.3 cos Exp[0.2 (0.25 arcsin(sin0.25 )

-10 -5 5 10

-10

-5

5

10

-1 -0.5 0.5 1

-1

-0.5

0.5

1

[ 0, 80] [ 0, 40]S u p e r m a t h e m a t i c s P a r a b l e s

2 4 6 8 10

-3

-2

-1

1

2

3

]1,1[),0],1,1[(, +=

=

=

sS

aexy

aexx


113/133

112

A R A B E S Q U E S 2

-6 -4 -2 2 4 6

-1

1

2

3

4

-6 -4 -2 2 4 6

-1

1

2

3

4


114/133

113

S u p e r m a t h e m a t i c s f u n c t i o n s cex xy and sex xy

cex xy = cos[xy-arcsin[s.sin (xy - ]] for s = 0.4 and s = 0.9 ;x [-3,3], y [-3,3

1 2 3 4 5 6 7

1

2

3

4

5

6

7

1 2 3 4 5 6 7

1

2

3

4

5

6

7

S (s = 0.4, = 0) S (s = 0.9, = 0)

sex xy = sin[xy-arcsin[s.sin (xy - ]] for s = 0.4 and s = 0.9 ;x [-3,3], y [-3,3

1 2 3 4 5 6 7

1

2

3

4

5

6

7

1 2 3 4 5 6 7

1

2

3

4

5

6

7

S (s = 0.4, = 0) S (s = 0.9, = 0)


115/133

selariu supermathematics functions, editor florentin smarandache

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