Projections of straight lines: a solved example
A line AB has its end A 15 mm below HP and 20 mm behind VP. The end B is 80 mm below HP and 60 mm behind VP. The front view is inclined at 50° to XY. Draw the projections and find the true length and the true inclinations with HP and VP.
Every line below was drawn and checked by Graspline's own geometry engine, by the textbook method, one step at a time with the reason for each.
The construction, step by step
Draw the reference line XY
XY is where the horizontal plane (HP) and the vertical plane (VP) meet. Every distance in the drawing is measured from it.
Front view (a′, b′): above XY when the point is above HP, below XY when it is below HP.
Top view (a, b): below XY when the point is in front of VP, above XY when it is behind VP.Locate the projections of A
A is 15 mm below HP, so a′ is 15 mm below XY. A is 20 mm behind VP, so a is 20 mm above XY.
Both projections lie on one vertical line: the projector of A.
Draw the loci of b′ and b
B is 80 mm below HP, so b′ must lie on a horizontal line 80 mm below XY — the locus of b′. B is 60 mm behind VP, so b lies on a line 60 mm above XY — the locus of b.
We know which lines b′ and b sit on, but not yet where along them.
Draw the front view at 50° to XY
The front view makes α = 50° with XY. From a′ draw a line at 50°, sloping downward to the right, until it meets the locus of b′. That point is b′.
a′b′ is the front view: 84.9 mm. Its horizontal run, 54.5 mm, is the distance between the two projectors.
Project b′ to get b, then draw the top view
Draw the projector through b′ across to the locus of b. Where they meet is b. Join ab: the top view, 67.6 mm, at β = 36.3° to XY.
Neither view shows the true length, because the line is inclined to both HP and VP. α and β are only apparent angles.
Turn the top view parallel to XY: true length and θ
Rotate the top view ab about a until it is parallel to XY: b swings along an arc to b₁ (ab₁ = ab). The line is now parallel to VP, so its front view shows the true length.
Project b₁ down to the locus of b′ to get b₁′: the distance of B from HP has not changed, so b₁′ must stay on that locus. Join a′b₁′: TL = 93.8 mm, and its angle with XY is θ = 43.9°, the true inclination with HP.
Turn the front view parallel to XY: φ
Now rotate the front view a′b′ about a′ until it is parallel to XY: b′ moves to b₂′. The line is now parallel to HP, so its top view shows the true length.
Project b₂′ up to the locus of b to get b₂. Join ab₂: 93.8 mm, the same TL as before, which confirms the construction. Its angle with XY is φ = 25.2°, the true inclination with VP.
Result
True length, TL93.8 mmTrue inclination with HP, θ43.9°True inclination with VP, φ25.2°Front view 84.9 mm at α = 50°; top view 67.6 mm at β = 36.3°; end projectors 54.5 mm apart.
✓ Verified: every stated value is read back from the drawing; θ + φ = 69.1° (never more than 90°).
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