I have a triangle in (u,v) coordinates in an image. I would like to draw this triangle at 3D coordinates (X,Y,Z) texture-mapped with the triangle in the image.

Here, u,v,X,Y,Z are all vectors with three elements representing the three corners of the triangle.

I have a very ugly, slow and insatisfactory solution in which I (1) extract a rectangular part of the image, (2) transform it to 3D space with the transformation defined by the three points, (3) draw it with surface, and (4) finally masking out everything that is not part of the triangle with AlphaData.

Surely there must be an easier way of doing this?