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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