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 ?

Definition:
If any line bisects the oppozite sides of a Quadrilateral A, B, C, D in points P and Q in such a manner the ratio AB:DC=AP:DQ holds than this line is called a Sweeping Line.

Theorem by Milan Anthony Vlasak:

"There is only one Parabola which is tangent to all Sweeping Lines of a Quadrilateral."

Any objection to that ?
You guys are mathematitions, I am a surveyor.

Milan