Bilinear interpolation on quadrilateral

Question

I am interpolating data from the vertices of a quadrilateral to any random point inside the quadrilateral. I implement this by first doing a coordinate transformation which reshapes the quadrilateral to a unit square and then using bilinear interpolation. Does any of the python libraries already have an implementation to do this?

Answer 1

Bilinear interpolation between the corners of a quadrilateral gives the coordinates

P = (P00 (1-u) + P10 u) (1-v) + (P01 (1-u) + P11 u) v
  = P00 + (P10-P00) u + (P01-P00) v + (P11-P10-P01+P00) uv.

This forms a system of two quadratic equations in the unknowns u, v. If P is inside the quadrilateral, u, v ε [0, 1].

If we take the dot and cross products with P11-P10-P01+P0, the system takes the form

A + B u + C v + D uv = 0
a + b u + c v = 0

from which we eliminate v by

c (A + B u) - (C + D u) (a + b u) = -b D u² + (B c - b C - a D) u + (A c - a C) = 0

which is quadratic univariate. Compute the roots and keep the one in the domain. From u you get v.