Website owner: James Miller
ORTHOGONAL PROJECTIONS OF PLANE AREA REGIONS
Theorem. Let a region Q in a plane A be projected orthogonally onto some other plane B to produce a region P. Then
Area of P = (area of Q) cos θ
where θ is the acute angle between the two planes. See Fig. 1.
If n and m are unit vectors perpendicular to planes A and B respectively, then
Area of P = | n·m | (area of Q)
where | n·m | is the absolute value of the dot product of n and m.
Proof. Prove that the area of the projection P of region Q is given by (area of Q) cos θ .
To prove this result consider Fig. 2 where we have planes A and B intersecting along line RS and making an angle of θ with each other. A rectangular region Q of dimensions a b in plane A is projected orthogonally onto region P of plane B. The area of Q is ab. The area of P is ac = ab cos θ. Thus the area of P is given by
Area of P = (area of Q) cos θ .
We can use this result to find the area of the orthogonal projection of any arbitrary region Q in plane A onto the plane B by partitioning Q into narrow rectangular slices that run perpendicular to the line of intersection RS of planes A and B as shown in Fig. 3. Each slice projects into an image whose area is given by
(area of slice) cos θ .
Summing all slices gives the final result.
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