Projecting a Hexagonal Prism Tilted About a Base Edge
A hexagonal prism of base side 25 mm and axis 65 mm is to be drawn with one base edge on HP, that base edge parallel to VP, and the axis inclined at 40° to HP. The candidate draws the final elevation first: a hexagon standing on a corner with its axis at 40° to xy, and then projects the plan from it. The resulting plan shows the top base as a regular hexagon of 25 mm side, and the axis measures 65 mm in the plan. The outline looks like a tilted prism, yet the teacher marks the construction wrong. The candidate argues that the elevation clearly shows the 40° tilt and the 65 mm axis, so the drawing should be acceptable. The same prism is later required with the base edge perpendicular to VP.
1. Explain why drawing the final elevation first produces a wrong plan, and outline the correct three-stage construction.
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Drawing the final elevation first fixes the tilt and the edge position simultaneously, and it uses a true hexagon in the elevation. A true hexagon appears in elevation only when the base is parallel to HP or VP; once the base is tilted 50° to HP, its elevation must be foreshortened. Projecting back from that false true-shape forces the plan to show a regular hexagon, which is impossible for an inclined base.
The correct sequence is:
- Simple position. Base on HP, axis perpendicular to HP, the chosen base edge parallel to VP. Draw the plan as a regular hexagon with that edge parallel to xy, then the elevation as a rectangle 65 mm high.
- Tilt about the base edge. In elevation, rotate the axis to 40° to xy while the base edge stays on HP. Project the new plan; the base now appears foreshortened.
- Final rotation if needed. If the base edge is not yet parallel to VP, rotate the plan about the vertical axis until it is, then project the final elevation.
2. Determine the true lengths of the axis in elevation and plan, and state whether the top base appears regular in plan.
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Because the base edge is parallel to VP and the axis is perpendicular to that edge, the axis lies in a plane perpendicular to VP. Hence the axis is parallel to VP, and the elevation shows its true length: 65 mm at 40° to xy. The plan length is the horizontal projection: 65 cos 40° = 49.8 mm, so the top base centre is 49.8 mm from the base edge in plan.
The top base does not appear regular. The base plane is inclined at 50° to HP, so the hexagon is foreshortened in the direction perpendicular to the base edge, while the dimension parallel to the base edge remains 25 mm. The base edge on HP projects true length, 25 mm. The apparent 65 mm axis in the candidate's plan is therefore a direct symptom of the wrong construction.
3. The same prism is required with the base edge perpendicular to VP instead. State what changes and what stays the same.
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The solid, the 40° tilt and the 65 mm true axis length stay the same. What changes is the starting orientation in the simple position: the chosen base edge is now drawn perpendicular to xy in the plan, and the final rotation stage is adjusted so that this edge ends up perpendicular to VP. Because the axis is then no longer parallel to VP, the elevation no longer shows the axis true length; both views foreshorten it, and the plan of the base edge also foreshortens. The lesson is that the three-stage method is unchanged, but the view in which true lengths appear depends on the chosen edge orientation, which must be decided before starting.