That is in 2D:

X = a + b*x + c*y + d*x*y
Y = e + f*x + g*y + h*x*y

After finding the eight parameters a,b,c,d,e,f,g,h for a (parallel with x,y, grids) Rectangle A,B,C,D to general and 'tilted' Quadrilateral A',B',C',D' the points inside and on the perimeter of the Rectangle transforms nicely following the Quadrilateral.

Why the transformation cannot be reversed ?