Trilinear interpolation for distorted cubes? Is there a variation of trilinear interpolation that works on a "cube" that has been distorted (by moving one or more of its corners by an arbitrary amount into an arbitrary direction)?

I know that I could use the inverse of the distance to each of the 8 corners as weights to get an interpolated point within the distorted cube, but that would involve taking a lot of square roots which would slow my algorithm way down. I like the elegant way of trilinear interpolation but I can't figure out how to adapt it to a distorted cube. Any hint on a fast interpolation method for this situation would be much appreciated.
Imagine a cube with each corner a different RGB color value - as you know you could interpolate the colors inside the cube using trilinear interpolation. 

You can reuse this same logic for coordinates as well -- set the RGB values of the corners to be the XYZ coords of your distorted cube's corners. 

Now the RGB "color" you sample in this cube is the distorted coordinate -- eg, it is the coordinate of your sampling point after that point has been "remapped" to the now-distorted cube, using the normal trilinear algorithm.

Hope this helps.