projections of lines

7/25/2019 Projections of Lines

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Projections of lines

Straight line

A line is a geometric primitive that has length and direction, but no thickness. Straight

line is the Locus of a point, which moves linearly. Straight line is also the shortest

distance between any two given points.

The location of a line in projection quadrants is described by specifying the distances of

its end points from the !, "! and !!. A line may be#

!arallel to both the planes.

!arallel to one plane and perpendicular to the other.

!arallel to one plane and inclined to the other.

$nclined to both the planes.

Projection of a line

The projection of a line can be obtained by projecting its end points on planes ofprojections and then connecting the points of projections. The projected length and

inclination of a line, can be di%erent compared to its true length and inclination.

Case 1. Line parallel to a plane

&hen a line is parallel to a plane, the projection of the line on to that plane will be its

true length. The projection of line ABlying parallel to the ertical plane '!( is shown in

)gure * as ab.


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+igure *. !rojection of line on !. Line A is parallel to !.

Case 2. Line inclined to a plane

&hen a line is parallel to one plane and inclined to the other, The projection of the line

on the plane to which it is parallel will show its true length. The projected length on the

plane to which it is inclined will always be shorter than the true length. $n )gure -, the

line A is parallel to ! and is inclined to "!. The angle of inclination of A with "! is

being degrees. !rojection of line A on ! is a/b/ and is the true length of A. The

projection of line A on "! is indicated as line ab. Length ab is shorter than the true

length A of the line.


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+igure -. !rojection of line A parallel to 0 and inclined to "!.

Case 3. Projection of a line parallel to both HP and VP

A line A having length 12 mm is parallel to both "! and !. The line is 32 mm above"!, 42 mm in front of !. 5nd is 62 mm in front of right !!. To draw the projection ofline A, assume the line in the )rst quadrant. The projection points of A on the verticalplane !, hori7ontal plane "! and 8ight !ro)le plane !! is shown in )gure 6'a(. Sincethe line is parallel to both "! and !, both the front view a9b9 and the top view ab are intrue lengths. Since the line is perpendicular to the right !!, the left side view of the linewill be a point a::'b::(. After projection on to the projection planes, the planes arerotated such that all the three projection planes lie in the same planes. The multi;viewdrawing of line A is shown in +igure 6'b(.


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+igure 6. !rojection of line parallel to both "! and !.

Case . Line perpendic!lar to HP " parallel to VP

A line A of length 12 mm is parallel to ! and perpendicular to "!. The line is 12 mmin front of ! and 12 mm in front of right !!. The lower end of the line is 62 mm above"!. The projections of line A shown in )gure < can be obtained by the followingmethod.

=raw a line >? which is the intersection between ! and "!. =raw the front view a9b9 @12 mm perpendicular to the >? line, with the lower end b9 lying 62 mm above the >?


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line. !roject the top view of the line which will be a point a'b( at a distance of 42 mmbelow >? line. Since the line is 32 mm in front of the right !! draw the > *?*line at adistance of 32 mm on the right; side of the front view.

Through 0 the point of intersection of >? and >*?*, lines draw a


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Line parallel to one plane and inclined to the other

Case #. Line parallel to VP and inclined to HP

A line A, C2 mm long is inclined at 62B to "! and parallel to !. The line is 12 mm infront of !. The lower end A is 62 mm above "!. The upper end is 2 mm in front ofthe right !!. The projections of line A shown in )gure can be obtained in thefollowing manner. Dark a9, the front view of the end A, 62 mm above "!. =raw the frontview a:b: @ C2 mm inclined at 62B to >? line.

!roject the top view abparallel to >? line. The top view is 12 mm in front of !. =rawthe >*?*line at a distance of 2 mm from b$. =raw a


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(b)

+igure . !rojections of line A parallel to ! and inclined to "!.

Case *. Line inclined to HP and VP

&hen a line is inclined to both "! and !, the apparent inclination of the line toboth the projection planes will be di%erent from the actual inclinations. Similarly theprojected length of the lines on to the planes will not be the same as the true lengthf the line. The following notation will be used for the inclinations and length of thelines for this entire lecture series#


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Actual inclinations are degrees to "! and F degrees to !.Apparent $nclinations are a and b to "! and ! respectively.

The Apparent Lengths of line A are ab and a:b:in the top view and front viewrespectively.

5Gample# =raw the projections of a line A inclined to both "! and !, whose truelength and true inclinations and locations of one of the end points, say A are given.

The projections of the line A are illustrated in )gure *. Since the line A is inclinedat to "! and F to ! H its top view ab and the front view a:b: are not in truelengths and they are also not inclined at angles to "! and F to ! in the +rontview and top view respectively. +igure - illustrates the projections of the line Awhen the line is rotated about A and made parallel to ! and "! respectively. Aclear understanding of these can be understood if the procedure followed in thesubsequent sub;sections are followed#

+igure *# The projections of a line inclined to both "! and !

Step 1: Rotate the line AB to make it parallel to VP.

8otate the line A about the end A, keeping , the inclination of A with "! constant tillit becomes parallel to !. This rotation of the line will bring the end to the newposition *. A*is the new position of the line A when it is inclined at q to "! andparallel to !. !roject A*on ! and "!. Since A*is parallel to !, a:b*:, the projection


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of A* on ! is in true length inclined at q to the >? line, and ab*, the projection ofA*on "! is parallel to the >? line. Iow the line is rotated back to its original positionA.

Step 2: Rotate the line AB to make it parallel to HP.

8otate the line A about the end A keeping F the inclination of A with ! constant, tillit becomes parallel to "! as shown in )gure -. This rotation of the line will bring theend to the second new !osition -. A-is the new position of the line A, when it isinclined at f to ! and parallel to "!.

!roject A-on "! and !. Since A-is parallel to "!, ab-, the projection of A-on "! is intrue length inclined at f to >? line, and a:b-: the projection of A-on ! is parallel to >?line. Iow the line is rotated back to its original position A.

Step 3: Locus of end B in the front ie!

8eferring to )gure -, when the line A is swept around about the end A by onecomplete rotation, while keeping the inclination of the line with the "! constant, the


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end will always be at the same vertical height above "!, and the locus of the end will be a circle which appears in the front view as a hori7ontal line passing through b9.

As long as the line is inclined at to "!, whatever may be the position of the line 'i.e.,whatever may be the inclination of the line with !( the length of the top view willalways be equal to ab* and in the front view the projection of the end lies on the locusline passing through b*/.

Thus ab*, the top view of the line when it is inclined at to "! and parallel to ! will beequal to ab and b:, the projection of the end in the front view will lie on the locus linepassing through b*:.

Step ": Locus of end B in the top ie!

$t is evident from )gure -, that when the line A is swept around about the end A by one

complete rotation, keeping f the inclination of the line with the ! constant, the end will always be at the same distance in front of ! and the locus of the end will be acircle which appears in the top view as a line, parallel to >?, passing through b.

As long as the line is inclined at F to !, whatever may be the position of the line 'i.e.,whatever may be the inclination of the line with "!(, the length of the front view willalways be equal to a9b-9 and in the top view the projection of the end lies on the locusline passing through b-.

Thus a:b-: the front view of the line when it is inclined at f to ! and parallel to "!, willbe equal to a9b9 and also b, the projection of the end in the top view lies on the locusline passing through b-.

Step #: $o o%tain the top and front ie!s of AB

+rom the above two cases of rotation it can be said that

'i(the length of the line A in top and front views will be equal to ab*and a9b-9respectively and

'ii( The projections of the end , 'i.e., b and bJ( should lie along the locus line passingthrough b-and b*: respectively.&ith center a, and radius ab -draw an arc to intersect the locus line through b -at b.

Eonnect ab the top view of the line A.Similarly with center a9, and radius a9b-9 draw an arc to intersect the locus line throughb*9 at b:. Eonnect a9b9 the front view of the line A.

%rthographic projections

As the location of one of the end points 'i.e. A( with respect to "! and !, is given,mark a anda:, the top and the front views of point A.$f the line A is assumed to be made parallel to ! and inclined at to "!. The front


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view of the line will be equal to the true length and true inclination of the line with "!.=raw a9b*9 passing through a9 at to >? line and equal to the true length of A. a9b*9 isprojected down to get ab*, the top view parallel to the >? line. This is illustrated in)gure 6.

+igure 6. $llustrates the true length and true inclination of the line when it is madeparallel to !.

Iow the line A is assumed to be made parallel to "! and inclined at F to !. This isshown in )gure ? line. The length ab-is equal to the truelength of A. The end points a and b-are projected on to a line parallel to >? line andpassing through a/ to get a9b-9 which is the front view of the line when it is parallel to "!and inclined to !. =raw the hori7ontal locus lines through b-, and b*9. &ith center a andradius ab*, draw an arc to cut the locus line drawn through b-at b. Eonnect ab, the topview of the line A. &ith center a9 and radius a9b -:, draw an arc to cut the locus linedrawn through b*9 at b9. Eonnect a9b9, the front view of the line A. 0rthographic


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projections of line A inclined to both ! and "!, illustrating the projected length, truelengths apparent inclinations and true inclinations are shown in )gure .

+igure


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+igure . $llustrates the true length, apparent lengths, tue inclination and apparentinclination of the line A inclined to "! and !..

+o ,ind +r!e length and tr!e inclinations of a lineDany times if thetop and front views of a line are given, the true length and trueinclinations of a line is required to be determined.

The top and front views of the object can be drawn from if any of the following data areavailable#'a( =istance between the end projectors,'b( =istance of one or both the end points from "! and ! and'c( Apparent inclinations of the line.

The problems may be solved by


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'i( 8otating line method or

'ii( 8otating trape7oidal plane method or

'iii( AuGiliary plane method.

-otating line &ethod

The method of obtaining the top and front views of a line, when its true length and trueinclinations are given.&hen a view of a line is parallel to the >? line, its other view will be in true length and attrue inclination.y following the procedure mentioned previously, in the reverse order, the true lengthand true inclinations of a line from the given set of top and front views can be found.The step by step procedure is shown below in )gure *.

+igure *. determinationof ture length and true inclinations of a line.

=raw the top view ab and the front view a9b9 as given


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8otation of the top view# &ith center a and radius abrotate the top view to the

new position ab* to make it parallel to the >? line. Since ab* is parallel to the >? line, its

corresponding front view will be in true length and at true inclination.

8otation of the front view# &ith center a9 and radius a9b9 rotate the front view to the new

position a9b-9 parallel to the >?line. Since a9b-J is parallel to the >? line, itscorresponding top view will be in true length and at true inclination. $n this position, theline will be parallel to "! and inclined at fto !. Through b draw the locus of in the topview. !roject b-9 to get b-, in the top view. Eonnect ab- +races of a line

The trace of a line is de)ned as a point at which the given line, if produced,

meets or intersects a plane.

&hen a line meets "!, 'or if necessary on the eGtended portion;of "!(, the point

at which the line meets or intersects the hori7ontal plane, is called hori7ontal trace

'"T(of the line and denoted by the letter ".

&hen a line meets ! 'or if necessary on the eGtended portion of !(, the point

at which the line meets or intersects the vertical plane, is called vertical trace 'T( of

the line and denoted by the letter .

&hen the line is parallel to both "! and !, there will be no traces on the said

planes. Therefore the traces of lines are determined in the following positions of the

lines.

+race of a line perpendic!lar to one plane and parallel to the otherSince the line is perpendicular to one plane and parallel to the other, the trace of theline is obtained only on the plane to which it is perpendicular, and no trace of the line isobtained on the other plane to which it is parallel. +igures - and 6 illustrates the traceof a line parallel tp2! and perpendicular to "! and parallel to "! and perpendicular to! respectively.


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+igure -. Trace of line parallel to ! and perpendicular to "!

+igure 6. Trace of a line perpendicular to the ! and parallel to

which will be in true length and true inclination f which the given line A makes

with !.

+races of a line inclined to one plane and parallel to the other

&hen the line is inclined to one plane and parallel to the other, the trace of the line isobtained only on the plane to which it is inclined, and no trace is obtained on the plane


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to which it is parallel. +igure < shows the hori7ontal trace of line A which is in lined "!and parallel to !

+igure < "ori7ontal trace of line A

+igure shows the vertical trace of line A which is inclined to ! and parallel to "!

+igure ertical trace of line A


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+races of a line inclined to both the planes

+igure 4 shows the ertical trace '( and "ori7ontal Trace '"( of Line A inclined at q to"! and K to !.

The line when eGtended intersects "! at ", the hori7ontal trace, but will never intersectthe portion of ! above >? line, i.e. within the portion of the ! in the * stquadrant.

Therefore ! is eGtended below "! such that when the line A is produced it will

intersect in the eGtended portion of ! at , the vertical trace.$n this case both hori7ontal trace '"( and ertical Trace '( of the line A lie below >?line.

+igure 4 ertical trace and hori7ontal trace of line A which is inclined to both verticalplane and hori7ontal plane.

projections of lines

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