How to map a point onto a warped grid

Question

Say you have a collection of points with coordinates on a Cartesian coordinate system.

Answer 1

The other answers are great. The only thing I'd add is that you might want to take a look at Free form deformation as a way of describing the deformations.

If that's useful, then it's quite possible to fit a deformation grid/lattice to your known pairs, and then you have a very fast method of deforming future points.