I have a quadrilateral that has known corner coordinates in two non-aligned 2-dimensional spaces (xy and pq). I have a point within the quadrilateral known in one space. How do I find it's coordinates in the other space?

Note that the spaces may not have coincident origin, sheer or rotation, and may be flipped and linearly distorted (that is the grid lines on one of the spaces may widen with respect to the other, but in a linear way).

I have tried a number of forms of interpolation only to find differing answers in the proof, and suspect the solution lies in creating a transformation matrix from the known points, which is beyond my current matrix math.

Here's my test case (note that pq of the quadrilateral may not be rectilinear as in this simplified test case):

Quadrilateral: ABCD

* A xy: (5,11), pq: (1,1)

* B xy: (10,5), pq: (0,1)

* C xy: (4,2), pq: (0,0)

* D xy: (1,7), pq: (1,0)

Point: M

* M xy: (3,6), pq: (?,?)

Using simple interpolation I soon found that I got different results depending on which line intersections I used as the linear distortion came into effect.

The second interpolation solution I tried was creating two bisectors of the quadrilateral passing through M that bisect AB, AD at the same ratio (r and 1-r) and BC, DC (s and 1-s), but I've not found a solution where I can determine both r and s.

Any experience in this area would be appreciated.

= Martin =