Graspline

Projections of planes (laminas): a solved example

A regular hexagonal lamina of 30 mm side rests on HP on one of its edges with its surface inclined at 45° to HP and perpendicular to VP. Draw its projections.

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.

XYabcdefa′(b′)f′(c′)e′(d′)a′(b′)f′(c′)e′(d′)θabcdef
The finished drawing.

The construction, step by step

  1. Draw the reference line XY

    Regular hexagonal lamina of 30 mm side. Its surface is inclined to HP, so we use the change-of-position method: first draw it lying in HP (true shape), then tilt it.

  2. Stage 1: draw the true shape in the top view

    Assume the lamina lies in HP. Its top view is then the true shape. Keep the resting edge ab perpendicular to VP (it appears as a line perpendicular to XY here and as a single point in the front view): that is the element the lamina tilts about.

  3. Stage 1: project the front view

    Every point is in HP, so the front view is a straight line on XY. Project each corner: corners on one projector share a point in this view, shown as a′(b′).

  4. Stage 2: tilt the front view to 45°

    Redraw the front view line turned about a′ (the resting edge ab stays on HP) until it makes θ = 45° with XY. The line is the same length: mark the points at the same distances along it.

  5. Stage 2: project the new top view

    Tilting about a line perpendicular to VP does not change any point's distance from VP. So draw horizontal loci from the stage-1 top view points; the projectors from the tilted line cut them at the new points. Join in order.

    This view is foreshortened: its area is the true area × cos θ.

  6. Result

    The final stage-2 views are the required projections.

    Highest corner above HP36.7 mm
    Width of final top view36.7 mm
    Length of the tilted front view line (stage 2)52 mm

    ✓ Verified: the foreshortened view has area = true area × cos θ (1653 = 2338 × cos 45°), the last stage is a rigid turn, and all edge lengths are preserved in 3D (max deviation 0 mm).

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