Let's say I have some coplanar points in R^3. I want to describe these points by coordinates on the plane they share, in R^2. This way I can do some tests faster. How should I do this?
Take any 3 distinct points A, B, and C. Take the cross product of (B - A) and (C - A) to get the normal, N, of the plane. Let (B - A) be one of your axes (normalize it). Take the cross product of N and (B - A) to get another orthogonal vector in the plane as the other axes (normalize it). Then project your points (relative to A) in your new axes.