True length and inclination of an oblique line in first-angle projection
Line AB is given in first-angle projection. End A is 20 mm above HP and 20 mm in front of VP; end B is 50 mm above HP and 60 mm in front of VP. The distance between the projectors of A and B measured along the reference line is 40 mm. The candidate draws the front view 50 mm long and the plan 56.6 mm long, then concludes that the true length is 56.6 mm because the plan is the longer view. The plan is placed above the front view. When asked for inclinations, the candidate measures the angle of the front view to the reference line and calls it the inclination with HP. No auxiliary view is drawn. The drawing is on A3 with HB pencil and 0.5 mm lines.
1. Calculate the true length and both inclinations. Show the three coordinate differences and explain why neither principal view gives the true length.
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Use the three coordinate differences: along the reference line Δx = 40 mm; height difference Δz = 50 - 20 = 30 mm; depth difference Δy = 60 - 20 = 40 mm.
True length TL = √(40² + 30² + 40²) = √4100 = 64.0 mm.
Inclination with HP: sin θ = Δz / TL = 30 / 64.0 = 0.469, so θ ≈ 27.9°. Inclination with VP: sin φ = Δy / TL = 40 / 64.0 = 0.625, so φ ≈ 38.7°.
The front view length is √(40² + 30²) = 50 mm and the plan is √(40² + 40²) = 56.6 mm. Each view omits one coordinate difference, so neither is true length. The plan is longer only because the depth difference (40 mm) is larger than the height difference (30 mm).
2. Correct the first-angle placement and explain how an auxiliary plane is used to obtain the true inclinations.
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In first-angle projection the top view is placed below the front view, not above it; projectors run vertically between them.
The front view shows Δx and Δz; the plan shows Δx and Δy. To get true length and inclination with HP, draw an auxiliary plane parallel to AB and perpendicular to HP, then project onto it. To get inclination with VP, draw a second auxiliary plane parallel to AB and perpendicular to VP. The angle of the foreshortened front view to the reference line is not the inclination with HP; it is only the angle of that view. The true angle must be measured in the auxiliary view that shows AB in true length.
3. State a quick numerical check that exposes the candidate's true-length error and give the corrected statement.
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A quick check is that true length cannot be shorter than any projection. The candidate's 56.6 mm is shorter than the computed 64.0 mm, so it cannot be the true length.
The corrected statement is: AB is about 64.0 mm long, inclined at about 27.9° to HP and about 38.7° to VP; both principal views are foreshortened.