How to find z by arbitrary x,y coordinates within triangle if you have triangle vertices - optimization

Question

Given vertices V1 (x1,y1,z1), V2 (x2,y2,z2), V3 (x3,y3,z3) of a triangle T, I have to find z coordinate of a point by it's x,y coordinate if I know that (x,y) lies within projection of triangle Tp (x1,y1), (x2,y2), (x3,y3).
Actually, triangle plane in 3D is defined by equation: Ax+By+Cz+D=0, and I can find z = (D-Ax-By)/C
The problem is that A, B, C, D are too expensive to calculate in run-time:
A = y1(z2-z3) + y2(z3-z1) + y3(z1-z2)
B = z1(x2-x3) + z2(x3-x1) + z3(x1-x2)
C = x1(y2-y3) + x2(y3-y1) + x3(y1-y2)
D = -x1(y2*z3 – y3*z2) – x2(y3*z1 – y1*z3) – x3 (y1*z2 – y2*z1)
Is it possible to calculate A, B, C, D using, say, opengl shaders? Are there optimized algorithms to find plane coefficients?

Answer 1

The technique is called Barycentric coordinates but the wiki page is pretty hard to follow -
See http://www.alecjacobson.com/weblog/?p=1596
float calcY(vec3 p1, vec3 p2, vec3 p3, float x, float z) {
float det = (p2.z - p3.z) * (p1.x - p3.x) + (p3.x - p2.x) * (p1.z - p3.z);
float l1 = ((p2.z - p3.z) * (x - p3.x) + (p3.x - p2.x) * (z - p3.z)) / det;
float l2 = ((p3.z - p1.z) * (x - p3.x) + (p1.x - p3.x) * (z - p3.z)) / det;
float l3 = 1.0f - l1 - l2;
return l1 * p1.y + l2 * p2.y + l3 * p3.y;
}
Code from http://www.gamedev.net/topic/597393-getting-the-height-of-a-point-on-a-triangle/ - carefull about computer graphics vs maths use of Y Z
ps. I Don't know of any faster version using shaders. One quick dirty+solution is to render the triangle using colors based on the height of the vertices and pick the pixel color at your X,Y - in practice this never ends up being much faster on a desktop machine, don't know about opengl-es