My question is simple but i can't remember the "transformation" formula:

I have a surface S in a 3d world. I know the coordinate of all four corners (P1, P2, P3, P4) where P1 is (x1, y1, z1) ect.

Then a random rotation and translation yielding the same surface is applied.

I found the new coordinate of all four corners (Q1, Q2, Q3, Q4).

I already found the transfomation matrix for all this 4 point so (R1, T1), (R2, T2) ...

But, How can i found the universal transformation matrix for all points of S? So, for example, i have coordinate of point P5 on our surface. How Can i found Q5?

Thanks

EDIT: I found the answer, was really stupid sorry. The transformation matrix are different for each point because it is composed by Rmat x tvec . Rmat is equal for all point but tvec no, it change depending by where the rotation was applied. BUT yeah, all point are in relationship, I can use only the trasformation matrix of one of this point, P1 and apply this formula; Q2 = (R1,t1) x P1 + P2* where P2* is P2 coordinate respect P1 = P2 - P1