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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.