A line AB, 70mm long, has its end A 15mm above HP and 20mm in front of VP.
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.
Answer: Front view a′b′ = 49.5 mm at 45° to XY, top view ab = 60.6 mm at about 54.7° to XY, end projectors 35 mm apart; HT is about 1.2 mm behind VP and VT about 0.9 mm above HP (B taken above HP and in front of VP).
Given: TL = 70 mm, A is 15 mm above HP and 20 mm in front of VP, \(\theta = 30^\circ\) (to HP), \(\phi = 45^\circ\) (to VP).
Useful lengths
- Rise of B above A: \(70\sin30^\circ = 35\) mm, so b′ is 50 mm above XY.
- Shift of B in front of A: \(70\sin45^\circ = 49.5\) mm, so b is 69.5 mm below XY.
- Top view length (depends on \(\theta\)): \(70\cos30^\circ = 60.6\) mm.
- Front view length (depends on \(\phi\)): \(70\cos45^\circ = 49.5\) mm.
- Distance between end projectors: \(\sqrt{49.5^2 - 35^2} = 35\) mm.
Construction
- Draw XY. Mark a′ 15 mm above XY and a 20 mm below XY on one projector.
- From a′ draw a′b₁′ = 70 mm at 30° to XY. Draw the locus of b′: a horizontal line through b₁′ (50 mm above XY). Project b₁′ down to the horizontal through a to get b₁; ab₁ = 60.6 mm is the top-view length.
- From a draw ab₂ = 70 mm at 45° to XY. Draw the locus of b: a horizontal line through b₂ (69.5 mm below XY). Project b₂ up to the horizontal through a′ to get b₂′; a′b₂′ = 49.5 mm is the front-view length.
- With centre a′ and radius a′b₂′ (49.5 mm) cut the locus of b′ at b′. With centre a and radius ab₁ (60.6 mm) cut the locus of b at b. Check that b′ and b lie on the same projector (35 mm from A's projector). Join a′b′ and ab (apparent angles: \(\beta = 45^\circ\), \(\delta \approx 54.7^\circ\)).
- HT: extend b′a′ to meet XY at h′ (15 mm to the left of a′'s projector). Project h′ down to meet ba extended: this point is HT, about 1.2 mm above XY, i.e. about 1.2 mm behind VP.
- VT: extend ba to meet XY at v (about 14.1 mm to the left of a). Project v up to meet b′a′ extended: this point is VT, about 0.9 mm above XY, i.e. about 0.9 mm above HP.
Both traces fall very close to XY because, extended backwards, the line passes almost through the XY line.