I am puzzled by something I just read in a paper and am trying to understand.
It's called "parametric trilinear transformation" and is basically a way by the author to transform the coordinates of a unit box into any skewed box. Problem is, all the author tells me other than that are these three equations:
X(s, t, u) = Ax * s + Bx * t + Cx * u + Dx * st + Ex * tu + Fx * su + Gx * stu + Hx
Y(s, t, u) = Ay * s + By * t + Cy * u + Dy * st + Ey * tu + Fy * su + Gy * stu + Hy
Z(s, t, u) = Az * s + Bz * t + Cz * u + Dz * st + Ez * tu + Fz * su + Gz * stu + Hz
where 0 \<= s, t, u \<= 1 are the coordinates in the unit cube and x, y, z are the real world coordinates after transforming into the skewed box.
There isn't even an explanation what A, B, ..., H are, though I very much suppose they are the cube's eight points in some way.
It's rather obvious that the vector P = A * s + B * t + C * u will be the point for any input s, t and u if we had only an unskewed box, as it simply means moving along the sides. The rest of the equations have to be taking care of the skewing. But that I cannot figure out.
Perhaps someone knows his stuff around this topic? I'm rather confused by the lack of information.