How to compute uv coordinates for an arbitrary quad?

Hi

For a regular grid, it is quite straightforward to get the uv coordinates for any given pixel in an image. However, for an arbitrary quad with vertices located at A,B,C,D, while A is the lower left vertex with uv coordinates (0,0) and C is the upper right vertex with uv coordinates (1,1). For any pixel inside the quad, how to compute the uv coordinates of the particular pixel? I found that a simple bilinear interpolation may not directly work.

Thank you very much!

Answer 1

[Julius Muschaweck]

With A == [Ax, Ay], etc, and a point P == [Px,Py] for certain u,v, the first step is indeed to subtract A from B, C, D and P, effectively placing A at the origin, i.e, Ax == Ay == 0, just like you do.

Then, you obtain P from u and v as

Px == u*(1 - v)*Bx + (1 - u)*v*Cx + u*v*Dx;

Py == u*(1 - v)*By + (1 - u)*v*Cy + u*v*Dy;

in standard degree 1 Bezier patch notation. The last term includes u*v, which complicates solving for u and v; the solution is indeed nonlinear in Px and Py.

You get

syms Bx By Cx Cy Dx Dy Px Py u v
[solu, solv] = solve([u*(1 - v)*Bx + (1 - u)*v*Cx + u*v*Dx - Px == 0, 
  u*(1 - v)*By + (1 - u)*v*Cy + u*v*Dy - Py == 0], [u, v])

solu = 

solv = 

which you can simplify using the 2D cross product, if you want, like

cross2D = @(AA,BB) AA(1)*BB(2) - AA(2)*BB(1);
syms B C D P
B = [Bx,By];
C = [Cx,Cy];
D = [Dx,Dy];
P = [Px,Py];
% and so on

and then e.g.

syms sigma_2
sigma_2 = 2 * cross2D(C,D-B)

sigma_2 = 

Such cross product are abundant in the solution.

You can then use these results to write a general function uv_from_P(A, B, C, D, P)