A projection is the transformation of points and lines in one plane onto another plane
by connecting corresponding points on the two planes with parallel
lines. This can be visualized as shining a (point) light source (located at infinity)
through a translucent sheet of paper and making an image of whatever is drawn on
it on a second sheet of paper. The branch of geometry dealing with the properties
and invariants of geometric figures under projection is called projective
geometry.
The projection of a vector onto a vector is given by
where is the dot
product, and the length of this projection is
General projections are considered by Foley and VanDam (1983).
The average projected area over all orientations of any ellipsoid is 1/4 the total surface area. This theorem also
holds for any convex solid.