Opengl calculate triangle derivatives

Question

I have a triangle in 2D space. I have screen space coordinates of each vertex, and I have attribute values of each vertex.

How can I calculate dFdx / dFdy for those attributes? In other words, how will change attribute from screen pixel to pixel.

//fragment shader
varrying vec2 myAttr;

void main(void)
{
  vec2 px = dFdx(myAttr);
  vec2 py = dFdy(myAttr);
}

I want to get px, py. I need to know delta(grow) of myAttr from next pixel for x and y axis. I need formula /algorithm how to calculate them manually (for example for cases when hardware does not support derivatives).

P.S. Attribute value linear interpolated between 3 vertices (according to OpenGL doc).

Answer 1

assuming your vertex are structured like this:

struct vertex
{
  double x; // screenspace x coordinate
  double y; // screenspace y coordinate
  double a; // your attribute
};

The derivation you're looking for are calculated like this:

      d    = (v0.x - v2.x) * (v1.y - v2.y) -
             (v1.x - v2.x) * (v0.y - v2.y);

      dfdx = ((v0.a - v2.a) * (v1.y - v2.y) -
              (v1.a - v2.a) * (v0.y - v2.y)) / d;

      dfdy = ((v0.a - v2.a) * (v1.x - v2.x) -
              (v1.a - v2.a) * (v0.x - v2.x)) / d;

Note that this equation becomes unstable as d approaches zero. It will also cause a divide by zero if d is exactly zero. This is not a problem in practice because in this case the triangle also has a area of zero and nothing needs to be rendered.