Line AB in Space: True Inclinations, Consistency Check and Error Diagnosis
A candidate is given the following data for a straight line AB: true length 70 mm, end A 15 mm above the HP and 20 mm in front of the VP, plan length 55 mm and elevation length 50 mm. The candidate must determine both true inclinations, locate end B, and satisfy himself that the data are consistent before starting the construction. On the sheet he ends up with a plan 75 mm long and an elevation drawn at 30° to XY, and writes that the line is inclined at 30° to the HP because the elevation shows 30°. The instructor wants the reasoning checked before any marks are awarded, and the client is also considering re-specifying the job as an inclination of 30° to the HP with the same true length and a plan of 55 mm.
1. Compute both true inclinations and the position of B, and test whether the four given data are consistent.
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Inclinations
- A view's length is the true length multiplied by the cosine of the inclination to the plane it is projected on.
- To HP: cos α = plan/TL = 55/70 = 0.7857 → α = 38.2°
- To VP: cos β = elevation/TL = 50/70 = 0.7143 → β = 44.4°
Position of B
- Height difference: Δz = √(TL² − plan²) = √(4900 − 3025) = √1875 = 43.3 mm. With A at 15 mm above HP, B is at 15 + 43.3 = 58.3 mm above HP (taking the end that rises).
- Distance from VP: Δy = √(TL² − elevation²) = √(4900 − 2500) = √2400 = 49.0 mm. With A 20 mm in front of the VP, B is at 20 + 49.0 = 69.0 mm in front.
Consistency test
- A view can never exceed the true length: 55 < 70 and 50 < 70 ✔
- sin²α + sin²β = 0.383 + 0.490 = 0.873 ≤ 1 ✔
- Cross-check: elevation from the coordinates = √(TL² − Δy²) = √(4900 − 2400) = 50.0 ✔
The data are consistent. Both angles obtained are true inclinations, because the plan and elevation are the orthogonal projections of the true-length line itself.
2. Diagnose the 75 mm plan and the claim that the line is inclined 30° to the HP because the elevation shows 30°.
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The 75 mm plan is impossible. No orthographic view of a line can be longer than its true length; the maximum is 70 mm, reached only when the line is parallel to that plane. A plan of 75 mm means one of the following has happened:
- The plan was drawn to the true length (70 mm) and then pacing/scale error added 5 mm - measurement error.
- The projectors were not drawn perpendicular to XY, so the plan is not an orthogonal projection at all.
- The plan was set out at the true inclination rather than the apparent one, i.e. the student measured along the line instead of onto the HP.
The 30° claim is a confusion of apparent and true inclination. The angle a line makes with XY in the elevation is its apparent inclination to the HP. It equals the true inclination only in the special case where the line is parallel to the VP. Here the line is inclined to both planes, so the elevation angle (30° as drawn) is not the inclination to the HP.
The true inclination is fixed by the lengths: cos α = 55/70 gives α = 38.2°, not 30°. The correct construction is the rotating-line (trapezoid) method or the auxiliary-plane method: draw the plan, erect a perpendicular at one end equal to the height difference, and measure the true angle there.
3. The client proposes re-specifying the job as 30° to the HP with TL 70 mm and plan 55 mm. Assess this proposal and state how the candidate should proceed.
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The proposal is self-contradictory. If the true length is 70 mm and the true inclination to the HP is 30°, the plan length is forced: plan = TL cos α = 70 × cos 30° = 70 × 0.866 = 60.6 mm, not 55 mm.
55 mm would correspond to α = cos⁻¹(55/70) = 38.2°. The two data sets describe different lines, so no construction can satisfy all three statements at once. This is exactly the kind of over-specified, inconsistent data that appears in drawings, and the correct response is to query it rather than to force a drawing to fit.
How to proceed
- Report the conflict with the numbers: 55 mm ↔ 38.2°, 30° ↔ 60.6 mm.
- Ask which value is controlling. In design, the functional requirement decides: if the angle is what matters (a ramp, a linkage), keep 30° and accept the plan of 60.6 mm. If the plan length is the constraint (clearance to another part), keep 55 mm and accept 38.2°.
- Once one pair is fixed, derive everything else: Δz = 70 sin 30° = 35 mm, so B is 35 mm above or below A; the inclination to the VP follows from the elevation length, and the position of B is completed in the plan and elevation with the usual projectors.
- Record the resolution as a revision note on the drawing so the shop works to one set of figures.