Projection of straight line Line inclined to both

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). PROBLEM End A of a line AB is 15 mm above HP & 20 mm in front of VP while its end B is 50 mm above HP and 75 mm in front of VP. The distance between end projectors of the line is 50 mm. Draw projections of the line and find its true length and true inclination with the principal planes. Also mark its traces.

b 1 ’ b’ θ: True inclination of the line with HP = 24º TL 50 a’ X HT VT’ h’ 15 θ α b 2 ’ α : Inclination of FV of the line with HP/XY Y v 50 20 Φ β b 1 a β : Inclination of TV of the line with VP/XY 75 TL b b Ø: True inclination of the line with VP = 41º 2

Type –II Line inclined to both HP & VP 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. PROBLEM A line AB, 70 mm long, has its end A 15 mm above HP and 20 mm in front of VP. It is inclined at 30° to HP and 45°to VP. Draw its projections and mark its traces.

b’ b 1 ’ 70 a’ 15 X 30° b 2 ’ HT VT’ h’ Y v 20 b 1 a 45° 70 b b 2

Q 10. 11 The top view of a 75 mm long line AB measures 65 mm, while its front view measures 50 mm. Its one end A is in HP and 12 mm in front of VP. Draw the projections of AB and determine its inclination with HP and VP To draw FV &TV of the line AB Given, TL=75 mm, TV=65 mm, FV=50 mm To find θ & Ø b 1 ’ b’ Ans. θ=31º 50 A is in HP & 12 mm→VP 75 Ans. Ø=49º a’ X Y 12 31º a Hint: Draw ab 1=65 mm // to XY. Because when TV is // to XY, FV gives TL. b 1 65 49º 75 65 b b 2

Q 10. 12 A line AB, 65 mm long has its end A 20 mm above H. P. and 25 mm in front of VP. The end B is 40 mm above H. P. and 65 mm in front of V. P. Draw the projections of AB and show its inclination with H. P. and V. P. Given, To draw FV &TV of the line AB A is 20 mm ↑ HP & 25 mm →V. P. B is 40 mm ↑ & 65 mm → V. P. To find θ & Ø b 1 ’ b’ 65 a’ b 2 ’ Hint 2: Draw locus of b’ 40 mm above XY & locus of b 65 mm below XY 40 TL=65 mm Hint 1: Mark a’ 20 mm above H. P & a 25 mm below XY 20 18º X 25 Y 38º Ans. θ=18º b 1 65 a 65 b b 2 Ans. Ø=38º

Q 10. 13: The projectors of the ends of a line AB are 5 cm apart. The end A is 2 cm above the H. P and 3 cm in front of V. P. The end B is 1 cm below H. P. and 4 cm behind the V. P. Determine the true length and traces of AB, and its inclination with the two planes Given, To find, True Length, θ, Ø, H. T. and V. T. A 0 B 0=50 mm A is 20 mm ↑ HP & 30 mm →V. P. B is 10 mm ↓ & 40 mm ← V. P. b b 2 20 HT X b 2’ 91 VT’ 40 a’ v h’ 10 50 30 b’ a 50º Y 20º Ans. θ=20º b 1 Ans. Ø=50º

Q 10. 14: A line AB, 90 mm 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 12 mm in front of V. P. Draw its front view and find its true inclination with the V. P. b’ Given, b 1 ’ T. L. =90 mm, θ=45º, β=60º A is in the H. P. & 12 mm→V. P. To find/draw, F. V. , T. V. & Ø 90 Ans. Ø = 38º a’ 12 X a Y 45º 60º 38º b 1 90 b b 2

Q 10. 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 65 mm. 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 Given, To find/draw, A 0 B 0=65 mm A is 25 mm ←V. P. & is ↓H. P. is 12 mm →V. P. & is above HP = 40º F. V. , T. L. , VT’ B θ b’ b 1 ’ b 2 ’ VT’ a HT b 2 20 X h’ a’ v 40º 12 25 b 1 b 65 Y

10. 17: A line AB, 90 mm long, is inclined at 30 to the HP. Its end A is 12 mm above the HP and 20 mm in front of the VP. Its FV measures 65 mm. Draw the TV of AB and determine its inclination with the VP 65 b’ b 1 ’ 90 a’ 12 Y 20 X 30° b 1 44° a Ans: Ø = 44º 90 b b 2

Q 10. 23: Two lines AB & AC make an angle of 120 between them in their FV & TV. AB is parallel to both the HP & VP. Determine the real angle between AB & AC. C c’ c 2’ c 1’ Ans. 112º 112° b’ 120° a’ X Y a b c 2 120° c c 1

Q 8: A line AB 65 mm long has its end A in the H. P. & 15 mm infront of the V. P. The end B is in the third quadrant. The line is inclined at 30 to the H. P. and at 60 to the V. P. Draw its projections.

Q 10. 19 A line AB, inclined at 40º to the V. P. has its end 50 mm and 20 mm above the H. P. the length of its front view is 65 mm and its V. T. is 10 mm above the H. P. determine. the true length of AB its inclination with the H. P. and its H. T. Given, To find, Ø = 40º, A is 20 mm↑HP, B is 50 mm ↑ HP, FV=65 mm, VT is TL, θ & HT ↑ HP b 1’ b’ 85 50 a’ 21º VT’ HT 10 b 2’ X v 40º b 1 a h’ 20 10 mm Y Ans, TL = 85 mm, = 21º & HT is 17 mm behind VP b 2 θ

Q 6. The top view of a 75 mm long line CD measures 50 mm. C is 50 mm in front of the VP & 15 mm 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. Given, To draw, TL = 75 mm, FV =50 mm, C is 15 mm ↓ HP & 50 mm → VP, D is 15 mm → VP FV & to find θ & Ø d 1’ d’ Hint 1: Cut anarc of 50 mm from c on locus of D 75 Hint 2: Make TV (cd), // to XY so that FV will give TL X 50 15 Y c’ c d 2 Ø=48º d Locus of D 50 θ=28º d 1 Ans: Ø=48º Ans: θ=28º

Q 10. 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, To draw, TL = 100, θ = 30º, Mid point M is 20 mm↑HP & in the VP End P in third quadrant & End Q in first quadrant FV & TV q’ p 2 X p 1’ p 50 m’ 30º 50 p’ 50 m q 2’ 20 p 2’ q 1’ q 2 50 q q 2 Y