True shape and dihedral angle for a triangular canopy panel
A design team is entering a canopy competition. The fabric panel is triangular, fixed to a vertical wall along AB and tensioned out to C. Using millimetre coordinates with x horizontal along the wall, y out from the wall and z vertical: A(0, 0, 400), B(1,200, 0, 400), C(600, 850, 1,100). The panel must be cut from a 1,250 mm wide roll of PVC-coated polyester, and the cutting pattern needs a 40 mm seam allowance on every edge. A student produced an auxiliary view by projecting the plan at 90° to line AC and measured the panel as 9% larger than the supplier’s estimate. The supplier’s estimate is based on true lengths. The CNC cutter needs a true-shape template, and the team also needs the dihedral angle between the panel and the wall to set the anchor plates. The student’s view also shows edge BC as 1,050 mm, which does not agree with the coordinates.
1. Explain why the student’s auxiliary view is not a true shape and outline the correct projection sequence.
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The student’s view is not a true shape because the auxiliary plane was not parallel to the plane ABC. Rotating the plan 90° to AC only makes AC appear perpendicular to the fold line; it does not produce an edge view of the whole plane. The correct sequence is: note that AB is horizontal and parallel to the vertical plane, so it appears true length in elevation; draw an auxiliary vertical plane perpendicular to AB in plan. A and B project to the same point in this view, so plane ABC appears as an edge; draw a second auxiliary plane parallel to that edge view. Project the three points perpendicular to the new fold line; the resulting triangle is the true shape. In the edge view, C lies 850 mm out from the wall and 700 mm above AB, so the edge view has a length of √(850² + 700²) = 1,101 mm. That is the true altitude of the triangle, not the plan altitude of 850 mm.
2. Calculate the true lengths and determine whether the panel with seam allowances fits the roll.
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True lengths: AB = 1,200 mm. AC = BC = √(600² + 850² + 700²) = √(360,000 + 722,500 + 490,000) = √1,572,500 = 1,254.0 mm. The true altitude from C to AB is √(850² + 700²) = 1,101.4 mm. Area = ½ × 1,200 × 1,101.4 = 660,840 mm², or 0.661 m². With a 40 mm seam on every edge, the width across the panel is 1,101.4 + 80 = 1,181.4 mm. The roll is 1,250 mm wide, so the true panel fits with 68.6 mm to spare. The student’s 9% oversize makes the altitude about 1,200 mm; adding seams gives 1,280 mm, which exceeds the 1,250 mm roll. That explains the supplier’s concern. The student’s BC value of 1,050 mm is also inconsistent with the coordinates and should have been a warning sign.
3. Determine the dihedral angle between the panel and the wall and explain how it should be set out.
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The dihedral angle is measured in a plane perpendicular to the fold line AB. Since AB runs along x, take a profile section at x = 600 mm. In that section the wall appears as a vertical line through A and B at y = 0, while the panel runs from (y = 0, z = 400) to (y = 850, z = 1,100). The panel therefore rises 700 mm while moving 850 mm out from the wall. The angle between the panel and the vertical wall is arctan(850/700) = 50.5°. The angle between the panel and the horizontal is arctan(700/850) = 39.5°. The anchor plate should be set at 50.5° to the wall, not at the plan angle. If the team used the plan angle, the anchor would be too flat and the fabric would be pulled out of plane. Mark the true angle on the edge view, not on the plan.