Let’s now define bilinear interpolation for a quadrilateral exactly the same way, except that instead of four texel colors, we’ll have the four vertices of the quad.

This sounds so ridiculously obvious when reading it that I would be surprised if nobody thought of working this out before. Or is the GPU code really difficult to work out compared to how simple the conceptual approach is?

By the way, the fact that bilinear interpolation creates quadratic splines along diagonals can be exploited to evaluate splines in a GPU texture unit.

That also sounds very interesting!