PROJECTION OF LINES

1. Projection of straight line
Line inclined to both HP & VP
Type-I
Given projections (FV & TV) of the line. To find True length & true
inclination of the line with HP (θ) and with VP(Φ).
End A of a line AB is 15mm above HP & 20mm in front of VP while
its end B is 50mm above HP and 75mm in front of VP. The distance
between end projectors of the line is 50mm. Draw projections of the
line and find its true length and true inclination with the principal
planes. Also mark its traces.
PROBLEM

2. a’
X Y
a
15
20
50
50
b’
75
b1
b1’
b2’
b2
θ α
Φ β
h’
HT
v
VT’
b
θ: True inclination of
the line with HP = 24º
α : Inclination of FV of
the line with HP/XY
Ø: True inclination of
the line with VP = 41º
β : Inclination of TV of
the line with VP/XY

3. Type –II
Given (i) T.L., θ and Ø,
(ii) T.L., F.V., T.V.
to draw projections, find α, β,H.T. and V.T.
Line inclined to both HP & VP
PROBLEM
A line AB, 70mm long, has its end A 15mm above HP and 20mm in front
of VP. It is inclined at 30° to HP and 45°to VP. Draw its projections and
mark its traces.

4. X
Y
15
a’
20
b1’
a
b2
30°
45°
b1
b
b’
b2’
h’
HT
v
VT’

5. Q10.11 The top view of a 75mm long line AB measures 65mm,while its front
view measures 50mm. Its one end A is in HP and12mm in front of VP. Draw the
projections of AB and determine its inclination with HP and VP
X Y
a’
12
a
b1
Hint: Draw ab1=65mm // to XY.
Because when TV is // to XY, FV
gives TL.
b1’
b’
65
b b2
Given,
TL=75mm,TV=65mm,FV=50mm
A is in HP & 12mm→VP
To draw FV &TV of the line
AB
To find θ & Ø
31º
49º
Ans. θ=31º
Ans. Ø=49º

6. Q10.12 A line AB, 65mm long has its end A 20mm above H.P. and 25mm in
front of VP. The end B is 40mm above H.P. and 65mm in front of V.P. Draw the
projections of AB and show its inclination with H.P. and V.P.
X Y
20
a’
25
a 40
65
Given,
TL=65mm
A is 20mm ↑ HP & 25mm →V.P.
B is 40mm ↑ & 65mm → V.P.
To draw FV &TV of the line
AB
To find θ & Ø
Hint1:Mark a’ 20mm above
H.P & a 25mm below XY
Hint2:Draw locus of b’ 40mm
above XY & locus of b 65 mm
below XY
b1’
b1
b
b’
b2’
b2
18º
38º Ans. θ=18º
Ans. Ø=38º

7. Q10.14:A line AB, 90mm long, is inclined at 45 to the H.P. and its top view
makes an angle of 60 with the V.P. The end A is in the H.P. and 12mm in front
of V.P. Draw its front view and find its true inclination with the V.P.
Y
X
a’
a
12
45º
b1’
b1
60º
b
b’
Given,
T.L.=90mm, θ=45º, β=60º A
is in the H.P. & 12mm→V.P.
To find/draw,
F.V.,T.V. & Ø
b2
38º
Ans. Ø = 38º

8. Q10.16:The end A of a line AB is 25 mm behind the V.P. and is below
the H.P. The end B is 12 mm in front of the VP and is above the HP The
distance between the projectors is 65mm. The line is inclined at 40 to
the HP and its HT is 20 mm behind the VP. Draw the projections of the
line and determine its true length and the VT
X Y
Given,
A0B0=65mm
A is 25mm ←V.P.& is ↓H.P. B
is 12mm →V.P. & is above HP θ
= 40º
To find/draw,
F.V., T.V., T.L., VT’
25
a
65 12
b
20
HT
h’
b1
40º
b1’
b’
a’
v
VT’
b2
b2’

9. 10.17:A line AB, 90mm long, is inclined at 30 to the HP. Its end A is 12mm above the HP and
20mm in front of the VP. Its FV measures 65mm. Draw the TV of AB and determine its
inclination with the VP
X Y
12
a’
20
a
30°
b1’
b’
b1
b b2
44°
Ans: Ø = 44º

10. Q6. The top view of a 75mm long line CD measures 50 mm. C is 50 mm in front of the VP &
15mm below the HP. D is 15 mm in front of the VP & is above the HP. Draw the FV of CD &
find its inclinations with the HP and the VP. Show also its traces.
X Y
15
50
Given,
TL = 75 mm, FV =50 mm,
C is 15mm ↓ HP & 50 mm → VP,
D is 15 mm → VP
To draw,
FV & to find θ & Ø
c’
c
Locus of D
Hint 1: Cut anarc of 50 mm
from c on locus of D
d
Hint 2: Make TV (cd), // to XY
so that FV will give TL
d1
d1’
d’
Ø=48º
Ans: Ø=48º
d2
θ=28º
Ans: θ=28º

11. Q10.10 A line PQ 100 mm long is inclined at 30º to the H.P. and at 45º to the V.P. Its
mid point is in the V.P. and 20 mm above the H.P. Draw its projections, if its end P is in
the third quadrant and Q is in the first quadrant.
Given,
TL = 100, θ = 30º, Mid point M is
20mm↑HP & in the VP
End P in third quadrant &
End Q in first quadrant
To draw,
FV & TV
X Y
20
m’
m
30º
q1’
p1’
q2
p2
q2’
q’
p’
q
p
q2
p1
p2’