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 ?

This is about a mapping function.
I can now formulate a new mapping function by this principle:

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

Sorry, there is a better Definition:

Edited Definition:
If any line INTERSECTS 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.