Projections of Projections of Straight LinesStraight Lines

Engineering GraphicsEngineering Graphics

TA 101TA 101

Projections of Straight LinesProjections of Straight Lines

The shortest distance between two points is The shortest distance between two points is called a called a straight line.straight line.

Positions of straight lines with respect to Positions of straight lines with respect to V.P. and H.P.V.P. and H.P.

1.1. Perpendicular to one plane and parallel to the other.Perpendicular to one plane and parallel to the other.

2.2. Parallel to both the planes.Parallel to both the planes.

3.3. Parallel to one plane and inclined to the other.Parallel to one plane and inclined to the other.

4.4. Contained by one plane and inclined to the other.Contained by one plane and inclined to the other.

5.5. Inclined, to both the planes.Inclined, to both the planes.

Line perpendicular to H.P. and Line perpendicular to H.P. and parallel to V.P.parallel to V.P.

Prob 1 :Prob 1 : A line AB 25 mm long is parallel A line AB 25 mm long is parallel to V.P. and perpendicular to H.P. Point A to V.P. and perpendicular to H.P. Point A is 35 mm above H.P. and 20mm in front of is 35 mm above H.P. and 20mm in front of V.P. Point B is 10mm above H.P. Draw the V.P. Point B is 10mm above H.P. Draw the projections of the line AB.projections of the line AB.

Prob 2 :Prob 2 : A line CD 20 mm long is parallel to V.P. A line CD 20 mm long is parallel to V.P. and and perpendicular to H.P. perpendicular to H.P. Point Point C is 35 mm above H.P. and 10 C is 35 mm above H.P. and 10

mm in front mm in front of of V.P. Draw its projections.V.P. Draw its projections.

Line perpendicular to Line perpendicular to V.P. V.P. and and parallel to parallel to H.P. H.P.

Prob 3 :Prob 3 : A line AB 25 mm long is A line AB 25 mm long is perpendicular to V.P. and parallel to H.P. perpendicular to V.P. and parallel to H.P. Its end A is 10mm in front Its end A is 10mm in front of of V.P. and the V.P. and the line is 20mm above H.P. Draw the line is 20mm above H.P. Draw the projections of the projections of the line.line.

Prob 4 : Prob 4 : A line CD 30 mm long is A line CD 30 mm long is perpendicular to V.P. perpendicular to V.P. and and parallel parallel to to H.P. H.P. Its end C is 5 mm, in front of V.P. and the Its end C is 5 mm, in front of V.P. and the line is 10 mm above' H.P. Draw the line is 10 mm above' H.P. Draw the projections of the line.projections of the line.

Line parallel to both the planesLine parallel to both the planes

Prob 5 :Prob 5 : A line CD 30 mm long is parallel A line CD 30 mm long is parallel to both the planes. The line is 40 mm to both the planes. The line is 40 mm above H.P. and 25 mm in front of V.P. above H.P. and 25 mm in front of V.P. Draw its projections.Draw its projections.

Prob 6 :Prob 6 : A line AB 50 mm long is parallel to both the planes. A line AB 50 mm long is parallel to both the planes. The line is 35 mm in front of V.P. and 30 mm above H.P. The line is 35 mm in front of V.P. and 30 mm above H.P.

Draw the projections of the lineDraw the projections of the line..

Line parallel to V.P. and inclined Line parallel to V.P. and inclined to H.P.to H.P.

Prob 7 :Prob 7 :A line PQ 40mm long is parallel to A line PQ 40mm long is parallel to V.P. and inclined V.P. and inclined at at an angle of 30° to H.P. an angle of 30° to H.P. The end P is 15 mm above H.P. and 20 The end P is 15 mm above H.P. and 20 mm in front of V.P. Draw the projections mm in front of V.P. Draw the projections ofof the linethe line

Prob 8 :Prob 8 : A line MN 50mm long is parallel to A line MN 50mm long is parallel to V.P. and inclined V.P. and inclined at at an angle of 30° to H.P. an angle of 30° to H.P. The end M is 20 mm above H.P. and 10 The end M is 20 mm above H.P. and 10 mm in front of V.P. Draw the projections of mm in front of V.P. Draw the projections of the linethe line

Line parallel to H.P and Inclined to V.PLine parallel to H.P and Inclined to V.P

Prob 9:Prob 9: Draw the projections of straight Draw the projections of straight line EF 40 mm long parallel to HP and line EF 40 mm long parallel to HP and inclined at an angle of 35inclined at an angle of 3500 to VP. The End to VP. The End E is 20 mm Above H.P and 15 mm in front E is 20 mm Above H.P and 15 mm in front of VP.of VP.

Prob 10:Prob 10: Draw the projections of a straight Draw the projections of a straight line CD 50mm long, parallel to H.P and line CD 50mm long, parallel to H.P and

inclined to V.P. the end C is 10 mm in front inclined to V.P. the end C is 10 mm in front of VP and D is 30 mm in front of VP. The of VP and D is 30 mm in front of VP. The

line is 15mm above HP.line is 15mm above HP.

Line contained Line contained by V.P. and by V.P. and inclined to H.P.inclined to H.P.

Prob 11:Prob 11: A line AB 50 mm long is in V.P. A line AB 50 mm long is in V.P. and inclined at an angle of 35° to H.P. and inclined at an angle of 35° to H.P. The The end A is 10 mm above H.P. Draw the end A is 10 mm above H.P. Draw the projections.projections.

Prob 12:Prob 12: A line A line EF 40 EF 40 mm long is in V.P. mm long is in V.P. and inclined to H.P. The top view and inclined to H.P. The top view measures 30 mm. The end measures 30 mm. The end E is E is 10 mm 10 mm above above H.P. Draw H.P. Draw the projections of the the projections of the line. Determine its inclination with H.P.line. Determine its inclination with H.P.

Line Inclined to Both the Planes:Line Inclined to Both the Planes:

The position of the line PQ which is The position of the line PQ which is inclined to both the planes is shown in first inclined to both the planes is shown in first quadrant in the Fig. quadrant in the Fig.

The projections of this line can be The projections of this line can be obtained by rotating this line into two obtained by rotating this line into two simple positions, namely,simple positions, namely,

a. parallel to the V.P.a. parallel to the V.P.

b. parallel to the H.P.b. parallel to the H.P.

15. 15. A line CD 80 mm long is inclined at an angle of 30A line CD 80 mm long is inclined at an angle of 3000 to to H.P and 45H.P and 4500 to V.P. the point C is 20 mm above H.P and 30 to V.P. the point C is 20 mm above H.P and 30 mm in front of V.P. Draw the projections of the straight line.mm in front of V.P. Draw the projections of the straight line.

Sol:Sol:1.1. C is 20 mm above H.P. therefore mark C’ 20 mm above XY C is 20 mm above H.P. therefore mark C’ 20 mm above XY

Plane.Plane.2.2. C is 30 mm in front of V.P. therefore mark C 30 mm below XY C is 30 mm in front of V.P. therefore mark C 30 mm below XY

Plane.Plane.To obtain the locus of d’ and top view length:To obtain the locus of d’ and top view length:3.3. To start with assume the line to be inclined to only one plane (say To start with assume the line to be inclined to only one plane (say

H.P). H.P). from C’ draw a 30from C’ draw a 3000 line to XY and mark d line to XY and mark d11’ such that C’d’ such that C’d11’=true ’=true

length = 80mm.length = 80mm.4.4. Draw a horizontal line through dDraw a horizontal line through d11’ which will be the locus of D in ’ which will be the locus of D in

front view.front view.5.5. Draw a horizontal line from C.Draw a horizontal line from C.6.6. From dFrom d11’ draw a projector to intersect the horizontal drawn from C ’ draw a projector to intersect the horizontal drawn from C

at dat d11. now cd. now cd11 is the length of the top view. is the length of the top view.

To To obtain obtain the locus of d and front view lengththe locus of d and front view length

7.7. Next assume the line to be parallel to H.P. and inclined Next assume the line to be parallel to H.P. and inclined at to V.P. We can obtain the locus of d.at to V.P. We can obtain the locus of d.

8.8. From c draw a line 45° to XY and mark dFrom c draw a line 45° to XY and mark d22 such that cd such that cd22= = true true length length = 80 mm.= 80 mm.

9.9. Draw a horizontal line through dDraw a horizontal line through d22 which- will be the locus which- will be the locus of D in top view.of D in top view.

10.10.From d2 draw a projector to intersect the horizontal line From d2 draw a projector to intersect the horizontal line drawn from c' at d'drawn from c' at d'22 . c'd'. c'd'22 is the length of the front view.is the length of the front view.

11.11.To draw the top view :To draw the top view :12.12.With c as centre and cd With c as centre and cd 11 as radius draw an arc to, as radius draw an arc to,

intersect the locus of d at d. Now cd is the top view. intersect the locus of d at d. Now cd is the top view. To To draw the front view :draw the front view :

13.13.With c' . as centre and c'd'With c' . as centre and c'd'22 as radius draw an arc to as radius draw an arc to intersect the locus of d'. Now c'd' is the front view. intersect the locus of d'. Now c'd' is the front view.

14.14.Now d' and d will be on the same projector.Now d' and d will be on the same projector.

17. A line PQ 75 mm long is inclined at an angle of 45° to 17. A line PQ 75 mm long is inclined at an angle of 45° to H.P. and 30° to V.P. The point P is 15 mm above H.P. and H.P. and 30° to V.P. The point P is 15 mm above H.P. and

20 mm in front of V.P. Draw the projections of the line.20 mm in front of V.P. Draw the projections of the line.

17. A line measuring 75 mm long has one of its ends 50 mm 17. A line measuring 75 mm long has one of its ends 50 mm in front of in front of V.P. V.P. and 15 mm above H.P. The top view of the and 15 mm above H.P. The top view of the line is 50 mm long. Draw line is 50 mm long. Draw and and measure the front view. The measure the front view. The

other end is 15 mm in front of V.P. and is above H.P.other end is 15 mm in front of V.P. and is above H.P.

1.1. A is 15 mm above H.P. and 50 mm in front of A is 15 mm above H.P. and 50 mm in front of V.P. Therefore mark a' 15 mm above XY and a V.P. Therefore mark a' 15 mm above XY and a 50 mm below XY.50 mm below XY.

To draw the top view, given the top view length :To draw the top view, given the top view length :

2.2. B is 15 mm in front of V.P. Therefore draw a B is 15 mm in front of V.P. Therefore draw a horizontal line 15 mm below XY to represent horizontal line 15 mm below XY to represent the locus of b.the locus of b.

3.3. The top view of the line is 50 mm. Therefore The top view of the line is 50 mm. Therefore with a as centre and 50 mm as radius, draw an with a as centre and 50 mm as radius, draw an arc to intersect the locus of b at b. Now ab is arc to intersect the locus of b at b. Now ab is the top view of the given straight line.the top view of the given straight line.

To obtain the locus of b'To obtain the locus of b'

4.4. With a as centre and ab as radius draw an arc With a as centre and ab as radius draw an arc to intersect the horizontal drawn through a at to intersect the horizontal drawn through a at b1.b1.

5.5. The true length is given as 75 mm. Therefore The true length is given as 75 mm. Therefore with a' as centre and 75 mm as radius, draw with a' as centre and 75 mm as radius, draw an arc to intersect the projector drawn from ban arc to intersect the projector drawn from b ll at b‘at b‘11..

6.6. Draw a horizontal line through b'1 to represent Draw a horizontal line through b'1 to represent the locus of b'.the locus of b'.

To draw the front viewTo draw the front view7.7. FromFrom bb draw a projector to intersect the draw a projector to intersect the

locus of b' at b'.locus of b' at b'.8.8. Join a'b' which is the front view of the given Join a'b' which is the front view of the given

line.line.

18. A line measuring 80 mm long has one of 18. A line measuring 80 mm long has one of its ends 60 mm above H.P. and 20 mm in its ends 60 mm above H.P. and 20 mm in front front of of V.P. The other end is 15 mm V.P. The other end is 15 mm above H.P. and in front of V.P. The front above H.P. and in front of V.P. The front view of the line is 60 mm long. Draw the view of the line is 60 mm long. Draw the top view.top view.

20. Draw the projections of a straight line AB of 20. Draw the projections of a straight line AB of 100mm long when one of its ends is touching the 100mm long when one of its ends is touching the V.P. and the other end touching H.P. the angle of V.P. and the other end touching H.P. the angle of

inclination with H.P and V.P are 40inclination with H.P and V.P are 4000 and 50 and 500 0

respectivelyrespectively

Sol:Sol:

To obtain the top view length and locus of b’To obtain the top view length and locus of b’

1.1. Mark any point a1’ on XY to represent the front Mark any point a1’ on XY to represent the front view of A in the above position.view of A in the above position.

2.2. Draw aDraw a11’b’b11’ = 100 mm at an angle of 40’ = 100 mm at an angle of 4000 to XY. to XY.

3.3. From bFrom b11’ draw the horizontal to represent the ’ draw the horizontal to represent the

locus of b’.locus of b’.

4.4. aa11bb11 is the corresponding top view length. is the corresponding top view length.

To obtain the front view length and locus of a:To obtain the front view length and locus of a:

5.5. Take another point bTake another point b22 on XY. Draw a line from b on XY. Draw a line from b22 at 50° at 50° ((ØØ ) to XY and mark a ) to XY and mark a22 bb22= 100 mm = true length.= 100 mm = true length.

6.6. Draw a horizontal through aDraw a horizontal through a22 to represent the locus of to represent the locus of a.a.

7.7. Draw the front view aDraw the front view a22 b b22 on XY corresponding to a on XY corresponding to a22

bb22 .Now a .Now a22 b' b'22 gives the length of the front view.gives the length of the front view.To obtain the final projections :To obtain the final projections :8.8. Mark some other point a' on XY. With a' as centre and Mark some other point a' on XY. With a' as centre and

aa22 b b22 as radius draw an arc to cut the locus of b' at b'. as radius draw an arc to cut the locus of b' at b'.9.9. From the configuration, a'b'From the configuration, a'b' b1b'1 . Therefore b1b'1 . Therefore

draw a'b' (front view) perpendicular to XY. Note that b draw a'b' (front view) perpendicular to XY. Note that b will coincide with a'.will coincide with a'.

10.10. With b as centre and aWith b as centre and all b bll as radius draw an arc to cut as radius draw an arc to cut the locus of a at a. Now ab is the top view.the locus of a at a. Now ab is the top view.

11.11. From the configuration, ab = a’From the configuration, ab = a’22 a a22. Hence ab is . Hence ab is perpendicular to XY.perpendicular to XY.

True Length of a Straight line and True Length of a Straight line and Its true Inclinations with H.P and Its true Inclinations with H.P and

V.P.V.P.Given the projections of a line, to find the Given the projections of a line, to find the true length and inclinations with H.P and true length and inclinations with H.P and V.P:V.P:when a line is parallel to a plane, its when a line is parallel to a plane, its projection on that plane will give its true projection on that plane will give its true length and the true inclination with the length and the true inclination with the other plane. After making the line parallel other plane. After making the line parallel to a plane, its true length can be obtained to a plane, its true length can be obtained by any one of the two methods.by any one of the two methods.

Rotating Line Method:Rotating Line Method:

Make each view parallel to XY line Make each view parallel to XY line and project the other view from it. and project the other view from it.

Trapezoid Method:Trapezoid Method:

Rotate the line about its projections Rotate the line about its projections till it lies in H.P. or V.P.till it lies in H.P. or V.P.

22. The distance between the projectors of two points A 22. The distance between the projectors of two points A and and B B is 70 mm. Point A is 10 mm above H.P. and 15 mm is 70 mm. Point A is 10 mm above H.P. and 15 mm in front of V.P. Point B is 50 mm above H.P. and 40 mm in in front of V.P. Point B is 50 mm above H.P. and 40 mm in

front of V.P. Find the shortest distance between A and front of V.P. Find the shortest distance between A and B by B by Rotating Line Method.Rotating Line Method. Measure the true inclinations of the Measure the true inclinations of the

line AB with V.P. and H.P.line AB with V.P. and H.P.

To obtain the projections :To obtain the projections :1. Mark a' 10 1. Mark a' 10 mm mm above XY and a 15 mm below above XY and a 15 mm below

XY. Draw another projector 70 mm from the XY. Draw another projector 70 mm from the projector" of A. Mark b' 50 mm above XY and b projector" of A. Mark b' 50 mm above XY and b 40 mm below XY on the second projector. Join 40 mm below XY on the second projector. Join a'b' and ab.a'b' and ab.

To obtain the shortest distance between A & To obtain the shortest distance between A & BB (= true-length (= true-length of of AB) :AB) :

2. Through b' draw the locus 2. Through b' draw the locus of of b'.b'. With a as centre and ab as radius draw an arc With a as centre and ab as radius draw an arc

to intersect the horizontal. line drawn from a at to intersect the horizontal. line drawn from a at bb22..

3. From b3. From b22 draw a projector to intersect the draw a projector to intersect the

locus of b' at blocus of b' at b22 . Join a'b . Join a'b22 . Now a'b' . Now a'b' 22= =

AB = shortest distance between A and B AB = shortest distance between A and B = 85 mm.= 85 mm.

CHECK .: Draw the locus of b. With a' as CHECK .: Draw the locus of b. With a' as centre and a'b' as radius draw an arc to centre and a'b' as radius draw an arc to intersect the horizontal line drawn from a' intersect the horizontal line drawn from a' at bat bII. From b', draw a projector to -meet . From b', draw a projector to -meet

the locus of b at bthe locus of b at b11. Join ab. Join abll. Now ab. Now ab11= =

AB = a'b’AB = a'b’22

23. The distance between the projectors of 23. The distance between the projectors of two ends of a straight line is 60 mm. One two ends of a straight line is 60 mm. One end is 15 mm above H.P. and 50 mm in end is 15 mm above H.P. and 50 mm in front of V.P. The other end is 60 mm front of V.P. The other end is 60 mm above H.P. and 10 mm in front of V.P. above H.P. and 10 mm in front of V.P. Draw the projections and find the true Draw the projections and find the true length of the line.length of the line.

24. The distance between the projectors of two points A 24. The distance between the projectors of two points A and and B B is 70 mm. Point A is 10 mm above H.P. and 15 mm is 70 mm. Point A is 10 mm above H.P. and 15 mm in front of V.P. Point B is 50 mm above H.P. and 40 mm in front of V.P. Point B is 50 mm above H.P. and 40 mm in front of V.P. Find the shortest distance between A and in front of V.P. Find the shortest distance between A and

B B by Trapezoid Methodby Trapezoid Method . . Measure the Measure the true inclinations of the line AB with V.P. and H.P.true inclinations of the line AB with V.P. and H.P.

To obtain the projections :To obtain the projections :1.1. Mark a' 10 Mark a' 10 mm mm above XY and a 15 mm below XY. above XY and a 15 mm below XY.

Draw another projector 70 mm from the projector of A. Draw another projector 70 mm from the projector of A. Mark b' 50 mm above XY and b 40 mm below XY on Mark b' 50 mm above XY and b 40 mm below XY on the second projector. Join a'b' and ab.the second projector. Join a'b' and ab.

To To obtain the shortest distance obtain the shortest distance (= true length (= true length of AB) :of AB) :2. At a' draw a perpendicular to a'b' and mark a' A = 15 2. At a' draw a perpendicular to a'b' and mark a' A = 15

mm = distance of a from XY.mm = distance of a from XY.3. Similarly draw 3. Similarly draw a a perpendicular to a'b' at b' and mark b' perpendicular to a'b' at b' and mark b'

B = 40 mm = distance of b from XY. Join AB. Now-AB B = 40 mm = distance of b from XY. Join AB. Now-AB = = 85 85 mm.mm.CHECK :CHECK : At a draw a perpendicular to ab and mark aA At a draw a perpendicular to ab and mark aA = 10 mm = distance of a' from XY. At b draw a = 10 mm = distance of a' from XY. At b draw a perpendicular to ab and mark b B = 50 mm = distance perpendicular to ab and mark b B = 50 mm = distance of b' from XY. Join AB. Measure AB (= 85 mm).of b' from XY. Join AB. Measure AB (= 85 mm).

The distance between the projectors of The distance between the projectors of two ends of a straight line is. 40 mm. One two ends of a straight line is. 40 mm. One end is 15 mm above H.P.and 10 mm in end is 15 mm above H.P.and 10 mm in front of V.P. The other end is 40 mm front of V.P. The other end is 40 mm above H.P. and 40 mm in front of V.P. above H.P. and 40 mm in front of V.P. Find the true length and the true Find the true length and the true inclinations of the line inclinations of the line by by (i) Rotating line (i) Rotating line method and (ii) Trapezoid method. method and (ii) Trapezoid method. CompareCompare

26. A line LM 70 mm long, has its end L 10 mm above H.P. 26. A line LM 70 mm long, has its end L 10 mm above H.P. and 15 mm in front of V.P. Its and 15 mm in front of V.P. Its top top view and frontview and front

view measures 60 mm and 40 mm respectively. Draw the view measures 60 mm and 40 mm respectively. Draw the projections projections of of the line and determine its inclinations with the line and determine its inclinations with

H.P. and V.P.H.P. and V.P.

LM 70 mm ; I' and l ; lm = 60 mm ; l'm' = 40 mm.LM 70 mm ; I' and l ; lm = 60 mm ; l'm' = 40 mm.1. Mark I' 10.mm above XY and l 15 mm below XY.1. Mark I' 10.mm above XY and l 15 mm below XY.To find To find θθ and 1'm' :and 1'm' : 2. From l draw a line parallel 2. From l draw a line parallel toto XY and mark lml =60 mm = XY and mark lml =60 mm =

top view length.top view length.NOTE : The front view corresponding to this top view will NOTE : The front view corresponding to this top view will

give the true length and 0.give the true length and 0.3. From ml draw a projector. With l' as centre and true 3. From ml draw a projector. With l' as centre and true

length = 70 mm as radius draw an arc to intersect the length = 70 mm as radius draw an arc to intersect the above projector at m‘above projector at m‘11. .

4. Join l'm'1. Now measure 4. Join l'm'1. Now measure θθ = 31°. = 31°.5. Draw the locus of m'. With l' as centre and front view 5. Draw the locus of m'. With l' as centre and front view

length = 40 mm as radius draw an arc to intersect the length = 40 mm as radius draw an arc to intersect the locus of m' at m'. Join I'm'. This is the front view of LM.locus of m' at m'. Join I'm'. This is the front view of LM.

To To find find ØØ and and lm :lm : 6. Draw l'm'6. Draw l'm'22 = 40 mm = front view length and = 40 mm = front view length and

parallel to XY.parallel to XY.NOTE: The top view corresponding to this front NOTE: The top view corresponding to this front view will give the true length and view will give the true length and ØØ).).

7. From m'7. From m'22 draw a projector. With 1 as centre and draw a projector. With 1 as centre and true length 70 mm as radius draw an arc to true length 70 mm as radius draw an arc to intersect the above projector at m intersect the above projector at m 22..

8. Join lm8. Join lm22 and measure and measure ØØ= 55°.= 55°.

9. Through m 9. Through m 2 2 draw the locus of m. With l as draw the locus of m. With l as centre and 60 mm (top view length) as radius centre and 60 mm (top view length) as radius draw an arc to cut the locus of m at m. Join lm.draw an arc to cut the locus of m at m. Join lm.