Isometric ellipses and an impossible perspective layout
An engineering office must supply an isometric view at 1:1 of a plate 90 × 60 × 20 mm with a Ø40 through bore normal to the 90 × 60 face and a Ø20 bore in the 90 × 20 side face, for a maintenance manual. A junior draughtsman lays out the three axonometric axes at 120° to one another with one axis vertical, steps off 90 mm along the 30° axis, and draws the Ø40 circle as an ellipse with a 40 mm major axis and a 28.3 mm minor axis. A second task is a two-point perspective of the same plate for a catalogue: the two vanishing points are placed 200 mm apart on an A3 sheet (420 × 297 mm), the station point is set 250 mm in front of the picture plane, and the plate is centred on the principal point.
1. Check the ellipse against the two accepted isometric conventions and identify the inconsistency.
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Two conventions are in use. In the true isometric projection the three axes are foreshortened by 0.816, and a circle of diameter D appears as an ellipse with major axis D (unchanged, because the major axis is parallel to the picture plane) and minor axis 0.577D. In the isometric drawing convention used in industry, the axes are stepped off at full length, which is the same picture enlarged by 1/0.816 = 1.22, so the ellipse becomes 1.22D × 0.707D.
The junior's layout mixes the two. He steps off 90 mm at full length, which is convention 2, but gives the ellipse a major axis of 40 mm, which is convention 1, while taking the minor axis of 28.3 mm, which is convention 2. The result is a bore that looks squashed along its major axis: the ellipse is too narrow by a factor of 1.22 in one direction only, so the part reads as an oval hole rather than as a circle seen obliquely.
2. Give the corrected values for both bores under the chosen convention, and state the orientation rule for the major axis.
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Keeping convention 2, which is what the 90 mm already stepped off commits him to: Ø40 gives major axis 1.22 × 40 = 48.8 mm and minor axis 0.707 × 40 = 28.3 mm; Ø20 gives 24.4 mm × 14.1 mm.
Orientation rule: the major axis of the ellipse is always perpendicular to that axonometric axis which is normal to the face containing the circle. The Ø40 bore lies in the horizontal 90 × 60 face, whose normal is the vertical axis, so its major axis is horizontal. The Ø20 bore lies in a vertical face whose normal is one of the 30° axes, so its major axis is perpendicular to that axis in the drawing - that is, along the other 30° direction.
If convention 1 were chosen instead, the axes would be stepped off at 0.816 (90 → 73.5 mm, 60 → 49 mm, 20 → 16.3 mm) with ellipse 40 × 23.1 mm. Both conventions give the same picture at different sizes; mixing them within one view is not acceptable.
3. Assess the perspective layout quantitatively and recommend a workable solution.
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For a right-angled corner the station point lies on the circle whose diameter is V1V2, so the eye distance d from the picture plane satisfies d ≤ V1V2/2 = 100 mm; equivalently V1V2 = d(tan θ1 + tan θ2) ≥ 2d. A station point 250 mm in front of the picture plane would demand V1V2 ≥ 500 mm, so the layout is geometrically impossible: the vanishing points would lie far outside the A3 sheet.
With d = 100 mm the half-angle of the cone of vision would be arctan(210/100) = 64°, far beyond the 30° limit, so the plate would be grossly distorted toward the edges of the sheet. To hold a half-angle of 30° the drawing must stay within ±100 × tan 30° = ±57.7 mm of the principal point, i.e., inside a picture about 115 mm wide. The plate is 90 mm wide, so it can be made to fit only if it is drawn small and placed close to the principal point; the alternatives are a larger sheet, or accepting a drawing at 1:2 that the reader is told is not to scale.