How can I project an arbitrary plane identified by 4 points onto a 2d plane?

Question

The issue we are trying to solve the issue of locating a point in two different representations of a plane. The first plane we have is rotated to create perspective; the se

Answer 1

This is a special case of finding mappings between quadrilaterals that preserve straight lines. These are generally called homographic or projective transforms. Here, one of the quads is a square, so this is a popular special case. You can google these terms ("quad to quad", etc) to find explanations and code, but here are some for you.

Perspective Transform Estimation

a gaming forum discussion

extracting a quadrilateral image to a rectangle

Projective Mappings for Image Warping by Paul Heckbert.

The math isn't particularly pleasant, but it isn't that hard either. You can also find some code from one of the above links.

Update

And this is one of my favorites: Computing a projective transformation