Barycentric Coordinates as Interpolants

The points P0, P1 and P2 are the vertices of the triangle. P is a point at which interpolation will be performed. For purposes of this document, point P0 has cartesian coordinates (x0,y0,w0), color coordinates (r0,g0,b0) and texture coordinates (u0,v0). Points P1 and P2 are similarly defined. The cartesian coordinates are defined in perspective space, i.e., they have been multiplied by a perspective matrix and perspective divided. The color and texture coordinates are defined in world space. Note that we use the w-coordinate instead of z. We use this convention because w represents eye space z, i.e., the z-coordinate by which eye space (X,Y) coordinates are divided to produce perspective space (x,y) coordinates. As we will see, w is used to undo the perspective division so that interpolation occurs in eye space, which is equivalent to interpolation in world space. Moreover, for depth priority calculations, w is preferred to perspective z when a fixed-point representation is used for range priority calculations.