1. PROJECTIONS OF
STRAIGHT LINES

2. PROJECTIONS OF STRAIGHT LINES
Definition of Straight line:
A straight line is the shortest distance between two points.
Top views of two end points of a straight line, when joined, give the top view of the straight line.
Front views of the two end points of a straight line, when joined, give the front view of the straight line.
Both the above projections are straight lines.

3. Orientation of Straight Line in Space
A line in space may be parallel, perpendicular or inclined to either the H.P. or V.P. or both.
It may be in one or both the reference Planes.
Line ends may be in different Quadrants.
Position of Straight Line in space can be fixed by various combinations of data like distance of its end points from reference planes, inclinations of the line with the reference planes, distance between end projectors of the line etc.

4. Notatioans used for Straight Line
True length of the line:
Denoted by Capital letters. e.g. AB=100 mm, means that true length of the line is 100 mm.
Front View Length:
Denoted by small letters. e.g. a’b’=70 mm, means that Front View Length is 70 mm.
Top View Length:
Denoted by small letters. e.g. ab=80 mm, means that Top View Length is 80 mm.
Inclination of True Length of Line with H.P.:
It is denoted by θ. e.g. Inclination of the line with H.P. (or Ground) is given as 30º means that θ = 30º.

5. Inclination of True Length of Line with V.P.:
It is denoted by Φ. e.g. Inclination of the line with V.P. is given as 40º means that Φ = 40º.
Inclination of Front View Length with XY :
It is denoted by α. e.g. Inclination of the Front View of the line with XY is given as 50º means that α = 50º.
Inclination of Top View Length with XY :
It is denoted by β. e.g. Inclination of the Top View of the line with XY is given as 30º means that β = 30º.
End Projector Distance:
It is the distance between two projectors passing through end points of F.V. & T.V. measured parallel to XY line.

6. Line in Different Positions with respect to H.P. & V.P.
CLASS A: Line perpendicular to (or in) one reference plane & hence parallel to both the other planes
Line perpendicular to P.P. & (hence) parallel
to both H.P. & V.P.
(2) Line perpendicular to V.P. & (hence) parallel
to both H.P. & P.P.
(3) Line perpendicular to H.P. & (hence) parallel
to both V.P. & P.P.

7. Line in Different Positions with respect to H.P. & V.P.
CLASS B: Line parallel to (or in) one reference plane & inclined to other two planes
Line parallel to ( or in) V.P. & inclined to H.P.
by .
(2) Line parallel to ( or in) H.P. & inclined to V.P.
by .
(3) Line parallel to ( or in) P.P. & inclined to H.P.
by  & V.P. by .

8. Line in Different Positions with respect to H.P. & V.P.
CLASS C: Line inclined to all three reference planes ( Oblique lines )
Line inclined to H.P. by , to V.P. by  and also inclined to profile plane.

9. Class A(1) : Line perpendicular to P. P
Class A(1) : Line perpendicular to P.P. & hence parallel to both the other planes Y
V.P.
P.P. a’ b’ b” a” . B A Y X
H.P. b a z x

10. Class A(1) : Line perpendicular to P. P
Class A(1) : Line perpendicular to P.P. & hence parallel to both the other planes
V.P. a b X Y a’ b’
H.P.

11. P.P.
a”, b” Y1 . a’ b’ a b X Y
V.P.
H.P.
Class A(1) : Line perpendicular to P.P. & hence parallel to both the other planes

12. a’ b’ a b
a”, b” X Y1 .
F.V.
L.H.S.V.
T.V. Y
Class A(1) : Line perpendicular to P.P. & hence parallel to both the other planes

13. Exercise 1 :- A Line AB, 50mm long is perpendicular to the profile plane. The end A is 20mm below H.P. , 30mm behind V.P. & 10mm to the left of P.P. Draw the projections of straight line AB (i.e. Front View & Top View).
Data given :- Y1
T.L. = 50mm 50
Profile Plane b . a .
(2)Point A
20 below H.P.
30mm Behind V.P.
T.V.=T.L. 30
Line is perpendicular
to P.P. 10 X Y
Line is 10mm left of
P.P. 20 . . b’ a’
F.V.=T.L.
Scale :- 1:1

14. Y
V.P.
a’, b’ . A B Y X
H.P. b a X
Class A(2):Line perpendicular to V.P. & (hence) parallel to both the other Planes
(i.e. H.P. & P.P.)

15. Class A(2):Line perpendicular to V. P
Class A(2):Line perpendicular to V.P. & (hence) parallel to both the other Planes
a’, b’ X Y
V.P. .
H.P. a b

16. Class A(2):Line perpendicular to V.P. & (hence)
parallel to both the other Planes
V.P.
H.P.
a’, b’ X Y a b
T.V.
F.V. .

17. Exercise 2 :- A Line ABC, 80mm long is perpendicular to V
Exercise 2 :- A Line ABC, 80mm long is perpendicular to V.P & 50mm below H.P. Point B, 20mm from A is on V.P. A is in 4th quadrant. Draw the projections of line ABC. .
Data given :- c
T.L. = 80mm
(2) AB = 20, BC = 60 60 .
(3) Point B is in V.P. Y X b
Line is 50mm below H.P. 20 . 50
- Point A is in 4th quadrant a .
a’,b’,c’
- Line is perpendicular to V.P.
Scale :- 1:1

18. Y
V.P. a’ b’ A B Y X
H.P.
a,b . X
Class A(3):Line perpendicular to H.P. & (hence) parallel to both the other Planes

19. Class A(3):Line perpendicular to H. P
Class A(3):Line perpendicular to H.P. & (hence) parallel to both the other Planes
V.P. a’ b’ Y X
a,b .
H.P.

20. Class A(3):Line perpendicular to H. P
Class A(3):Line perpendicular to H.P. & (hence) parallel to both the other Planes X Y a’ b’
a, b .
H.P.
V.P.

21. . a,b 30 X Y a’ 50 F.V.=T.L. b’ Scale :- 1:1
Exercise 3:- A Line AB, 50mm long is perpendicular to H.P. & it is below H.P. Point A is on H.P. & 30mm behind V.P. Draw the projections of the line AB.
Data given :- .
a,b
T.L. = 50mm 30
(2) Point A
On H.P.
30mm Behind
V.P. . X Y a’
(3) Line is perpendicular
to H.P. 50
F.V.=T.L. b’ .
Scale :- 1:1

22. Class B(1): Line contained by ( or parallel to) V. P. & inclined to H
Class B(1): Line contained by ( or parallel to) V.P. & inclined to H.P. by  Y
V.P. A B a’ b’ X Y a b θ
H.P. X

23. Class B(1): Line contained by ( or parallel to) V. P. & inclined to H
Class B(1): Line contained by ( or parallel to) V.P. & inclined to H.P. by  Y X
V.P. b’ a’ a b θ
H.P.

24. Class B(1): Line contained by ( or parallel to) V. P. & inclined to H
Class B(1): Line contained by ( or parallel to) V.P. & inclined to H.P. by 
V.P. b’ a’ b a X Y θ

25. Exercise 4 :- A Line AB, 75mm long, is in V. P
Exercise 4 :- A Line AB, 75mm long, is in V.P. It makes an angle of 30º with the H.P. Point A is 20mm above H.P. Draw the projections of line AB. b’
Data given :-
F.V.=T.L.
T.L. = 75mm
(2)  = 30º .
==30º a’
(3)Point A = 20mm above H.P. 20 . a b X Y
T.V.
- Line AB is in V.P.
Scale :- 1:1

26. Class B(2) : Line parallel to (or contained by) H. P. & inclined to V
Class B(2) : Line parallel to (or contained by) H.P. & inclined to V.P. by  Y
V.P.
V.P. a’ b’ b’ a’ A B ø X Y
H.P. Y X
H.P.  a b a
= X b

27. Exercise 5 :- A Line AB, 120mm long, is parallel to H. P
Exercise 5 :- A Line AB, 120mm long, is parallel to H.P. and inclined to V.P. by 50º. Point B is 10mm above H.P. and 40mm on in front of V.P. Point A is behind V.P. Draw the projection of line AB.
Data given :- . a
T.L. = 120mm . . a’
F.V. b’
(2)  = 50º 10
(3) Point B
10 above H.P.
40mm in Front of V.P. X Y
=
P.L. = T.L 40
- Line is parallel to H.P. .
Point A is behind V.P. b
Scale :- 1:1

28. Class B(3): Line parallel to (or contained by) P.P., inclined to H.P. by  & to V.P. by X 
H.P.  b” b’  a b X Z

29. Class B(3): Line parallel to (or contained by) P.P., inclined to H.P. by  & to V.P. by X Y
H.P. a b

30. Exercise 6 :- The distance between the end projectors of line MN is zero. Point M is 40 mm below H.P. & 25 mm behind V.P. Point N is 15 mm below H.P. & 65 mm behind V.P. Draw its projections and find the angle of the line with H.P. and V.P. Also find the true length of the line. .
Data given :-
(1) Point M
40 below H.P.
25 mm behind V.P. n
T.V. . m
(2) Point N
15 below H.P.
65 mm behind V.P. 65 25 Y X
45
(3) End projector dist. = 0 . . 15 n’ n” 40 
F.V.  . .
T.L. m’ m”

31. Class C:Line inclined to H. P. by  & V. P. by  ( i. e
Class C:Line inclined to H.P. by  & V.P. by  ( i.e. Line inclined to both the planes) Y
V.P. a’ b’ B A X Y 
H.P.  a b X

32. Class C:Line inclined to H. P. by  & V. P. by  ( i. e
Class C:Line inclined to H.P. by  & V.P. by  ( i.e. Line inclined to both the planes) X Y
V.P.  a’ b’ a b 
H.P.

33. Class C:Line inclined to H. P. by  & V. P. by  ( i. e
Class C:Line inclined to H.P. by  & V.P. by  ( i.e. Line inclined to both the planes) X Y a  b b’
H.P.
V.P. a’ 

34. Exercise 7:- A Line AB, 90 mm long, is inclined to H. P
Exercise 7:- A Line AB, 90 mm long, is inclined to H.P. by 30° and inclined to V.P. by 45º. The line is in first quadrant with Point A 15 mm above H.P. and 25 mm in front of V.P. Draw the projection of line AB.
Locus of b’ b’
b1’
Data Given :-
(1) T.L.=90 mm
F.V.
T.L.= 90
(2) Θ =30° . α a’ θ
(3) Φ =45° 15
b2’
(4) Point A
15 above H.P.
25 mm in Front of V.P. Y X . 25 b1 a Φ
Answers :- β
(1) F.V.= 64 mm
T.V.
(2) T.V = 78 mm
Locus of b b2
(3)  = 45°
(4)  = 55° b

35. . . . . Data given :- Answers :-
Exercise 8 :- The distance between the end projectors of a straight line AB is 80mm. Point A is 10mm above H.P. and 30mm in front of V.P. Point B is 40mm above H.P. and 50mm behind V.P. Draw the projections and find the inclination of straight line AB with H.P & V.P. and the true length of the line.
Data given :-
E.P.D. = 80mm
(2) Point A
10 above H.P.
30mm in front of V.P.
Locus of b .
Locus of b’ b . b1
(3) Point B
40 above H.P.
50mm behind V.P. b’
b2’
F.V. . 50
b1’ 40 a’
Answers :-  10
(1)  =
15º X Y
T.V. 30 . 
(2)  =
43º a b2
(3) T.L. =
117mm 80

36. Exersice 9 : A room is 5m X 4. 5m X 4 m high
Exersice 9 : A room is 5m X 4.5m X 4 m high. Determine by method of projections of straight lines, distance between diagonally(solid) opposite corners of the room.
Locus of b’ b’ a’
Data Given :-
b2’
(1) Length of the room=L=5m H
(2) Breadth of the room=B=4.5m θ
(3) Height of the room=H=4m Y X a b b2
Answer :-
Diagonal distance between opposite corners of the room B
a’b2’= 7.826m L
Scale :- 1:100

37. - AB is perpendicular to P.P. - Legs are unequal (AB > AC)
Exercise 10 :- Two unequal legs AB and AC, hinged at A make an angle of 135º between them in their elevation and plan. Leg AB is perpendicular to the Profile Plane. Determine the real angle between them.
Data Given :-
(1)  = 135º
Locus of c’ C
(2)  = 135º
T.L. c’
T.L.
- AB is perpendicular to P.P.
c3’
c2’
- Legs are unequal (AB > AC) 
T.L.
Answer :-
B,b’
A,a’
- The real angle between two unequal legs = Y X c3
BAC=125º a b  c2 c
Locus of c
Scale :- 1:1

38. Exercise 11 :- Two Mangoes on a tree, planted near the compound wall of a bunglow, are 1m and 1.25m above the ground and 0.5m & 0.75m from a 15cm thick compound wall but on the opposite sides of it. The distance between Mangoes measured along the ground and parallel to the wall is 1m. Determine the real distance between centres of two mangoes.
Data Given :- p q2
(1) E.P.D. = 1m
(2) Point P
1m above ground
0.5 behind wall
0.5m
15cm
T.V. Y X
Locus of q’
(3) Point Q
1.25m above ground
0.75m in front of wall q’
q2’
F.V.
0.75m p’
(4) Wall thickness = 15cm q
Scale :- 1:20
Locus of q
1.25m
Answer :- 1m
- the real distance between centres of two mangoes = G L 1m
p’q2’= 1.63m

39. Exercise 12 : The F. V. of a line MN, 90 mm long, measures 65 mm
Exercise 12 : The F.V. of a line MN, 90 mm long, measures 65 mm. Point M is in V.P. and 20 mm below H.P. Point N is in the first quadrant. Draw the projections and find inclinations of line with H.P. and V.P.
Data Given:
Locus of n’
(1) Point M
20mm below H.P.
In V.P. n’
n1’
(2) T.L.= 90 mm .
F.V.
(3) F.V.= 65 mm m n1 X Y
(4) α = 45° Φ 20
(5) Point N is in first Quadrant . α θ m’
n2’
Answers:
T.V.
(1) Θ = 31°
(2) Φ = 44° n2 n
Scale:- 1:1
Locus of n

40. TRACES OF A LINE
Definition: When a line is inclined to a plane, it will meet that plane, produced if necessary. The point where the line or line produced meets the plane is called trace.
Horizontal Trace: The point of intersection of the inclined line with the H.P. is called Horizontal Trace or simply H.T.
Vertical Trace: The point of intersection of the inclined line with the V.P. is called Vertical Trace or simply V.T.

41. . . b’ Y B a’ A b a X Example to illustrate the concept of traces V.P.
F.V. B A 
H.P. . h  a b v
T.V.
H.T. .
V.T.

42. IMPORTANT POINTS REGARDING TRACES OF A LINE
- If a line is inclined to both H.P. & V.P. then its Front view, h’ and V.T. must be on the same straight line.
e.g. if front view of a line AB is a’b’, then h,a’,b’ and V.T. must be on a same straight line.
- If a line is inclined to both H.P. & V.P. then its Top view, v and H.T. must be on the same straight line.
e.g. if Top View of a line AB is ab, then v, a, b and H.T. must be on a same straight line.

43. IMPORTANT POINTS REGARDING TRACES OF A LINE
(1) If a line is parallel to any of the plane, it has no trace upon that plane.
e.g. If the line is parallel to horizontal plane then that line will not meet H.P and hence there will be no H.T. and only V.T.
V.P.
H.P. A B b a
a’, b’ .
V.T. Y

44. IMPORTANT POINTS REGARDING TRACES OF A LINE
(1) If a line is parallel to any of the plane, it has no trace upon that plane.
e.g. If the line is parallel to horizontal plane then that line will not meet H.P and hence there will be no H.T. and only V.T.
V.T. A B a’ a b b’ ø  .

45. IMPORTANT POINTS REGARDING TRACES OF A LINE
e.g. If the line is parallel to Vertical Plane then that line will not meet V.P and hence there will be no V.T. and only H.T. Y X
H.P.
a,b .
V.P. A B a’ b’
H.T.

46. IMPORTANT POINTS REGARDING TRACES OF A LINE
e.g. If the line is parallel to Vertical Plane then that line will not meet V.P and hence there will be no V.T. and only H.T. 20
=  =30º b’ a’ b a
F.V.=T.L.
T.V. h’
H.T.

47. Exercise 13 : A line AB, 80 mm long is seen as a straight line of length 55mm in its front view and of length 65 mm in its top view. Its end A is 10 mm above H.P. & is in first quadrant where as end B is 25 mm behind V.P. and is in Second Quadrant. Draw its projections and find out its inclinations with H.P. & V.P. and also locate its traces.

48. . . Data Given: (1) T.L.=80 mm (2) F.V. = 55mm (3) T.V. = 65mm
(4) End A
10 mm above H.P.
??? I.F.O V.P.
Locus of b’
(5) End B
??? above H.P.
25mm behind V.P.
b1’ b’ .
V.T.
Locus of b
F.V. b b2
Answers:
(1)  = 36°  a’ 25 10
b2’
(2)  = 46° X Y h v
(3) H.T.=46mm I.F.O. V.P.
T.V.  a
(4) V.T.=36mm above H.P. . 65 b1
H.T.

49. Exercise 14 : The end projectors distance of a line MN is zero
Exercise 14 : The end projectors distance of a line MN is zero. Its end M is 25mm below H.P. & 40mm behind V.P. where as end N is 10 above H.P. & 55mm in front of V.P. Draw its projections and find out its inclinations with H.P. & V.P. and also locate its traces.

50. . . . . . . . . . . Data Given: (3) E.P.D. = 0 Answers: (1)  = 20°
(1) End M
25 below H.P.
40 behind V.P.
(2) End N
10 above H.P.
55mm I.F.O.V.P. Z .
(3) E.P.D. = 0 m
Answers: . . 40
(1)  = 20° n’ . n” 10
(2)  = 70° . . X Y v
V.T. 
H.T.
(3) T.L.= 101mm 25 . . . m’ m”
(4) H.T.=27mm I.F.O. V.P. 55 h . n
(5) V.T.=12mm below H.P.
(6)  +  = 90°

51. Exercise 15 : A divider instrument of a compass box having two equal arms AB & BC hinged at B is kept in H.P. on its needle point A & C with the line joining A & C is perpendicular to V.P. It is seen in front view as a straight line 100mm long inclined at 30° to H.P. while it is seen in top view as an angle abc with <abc = 60°. Draw its front view and top view and find;
The height of point B above H.P.
The apart distance between the needle points
A & C
(3)The lengths of arms AB & BC with real
between them

52. f Locus of b’ Data Given: (1) F.V.=100 mm (2)  = 30°H
(3) <abc =2 = 60° 
a’,c’ X Y
b1’
(4) ac is perpendicular to V.P.
a,A
Answers:
(1) AB = 112mm 2
AB dist.
60° b
(2) 2 = 53°
b1, B
Locus of b’
(3) AC = 100 mm
c,C

53. Exercise 16 : The end A of a straight line AB, 120mm long, is 50 mm behind V.P. & 35 mm below H.P. The line is inclined to H.P. by 30° & has a point C on it in both the reference planes. Draw the projections of the line and find out its inclinations with V.P. Also locate its traces.

54. . Data Given: (2) T.L.=120 mm (3)  = 30° 
(1) End A
35 below H.P.
50 behind V.P.
(2) T.L.=120 mm a c1 b1 .
(3)  = 30° 
Locus of b’
(4) C on AB in both ref. Planes
T.V. of AC b’
b1’ 50 .
C,c’,c
c1’
Answers: X Y
H.T. & V.T.
(1) = 45°
F.V. of AC 35
(2) H.T.= in V.P.& H.P.  . a’ b b2
Locus of b’
b2’
(3) V.T.= in V.P. & H.P.

55. PROJECTIONS OF STRAIGHT LINES