Preserving shapes in perspective projection

Question

Perspective projection is very simple to perform, but when I tried to prove that certain geometric elements preserve their identity when projected I faced many difficulties though intuitively it seems obvious. I managed to prove that a 3D line is projected on a 2D line, but I can't prove for example that a half-plane is projected on a half-plane. Any ideas on the course of actions I could follow?

Answer 1

It depends on how exactly you define perspective projection, i.e. which properties are part of the definition and which are consequences thereof.

But you can think about this geometrically. You have a line in space, and an eye point. Assuming that the line didn't pass through the eye, connecting the eye to every point on the line yields a bundle of lines, which together form a plane. Now wherever you place your image plane in space, it will intersect that plane in a single line. Unless, of course, it is parallel to that plane, in which case the line lies “at infinity” from the 2d viewpoint.

Now to the half plane. Exclude the situation where the eye lies in the same plane as the half plane. Then connecting the eye to every point of the half plane will not yield a half space. Instead, you get two quarter spaces, one in front of the eye and one behind. A generic plane will intersect this in two distinct regions, so the image will not be a half plane. If you try to fix that by discarding the part of the world behind the eye, then your original statement of line maps to line becomes false, so that's no solution.