by C.G. Scoop Sept.,
2022

Last month, shortly after the Journal of
Computer Graphics Techniques published Barycentric
Quad Rasterization, Paul Haeberli asked whether the barycentric
method, which reduces distortion and discontinuity in a texturemapped
sphere, could also improve the rendering of a cone? Yes, it can.
At the tip of a cone, as at a pole of a sphere, a quad has a degenerate
edge defined by two coincident points (see Figure 6 of the paper). With
triangle rasterization, one of the points is ignored (see Figure 7). For
the sphere, the two points differ in uvcoordinates and thus triangle
rasterization produces a texture discontinuity. For the cone, however, the
two points also differ in surface normal, producing a shading
discontinuity as well. This problem is discussed by Eric Haines in his
2014 post, Limits
of Triangles.
The discontinuities in texture can be overcome by avoiding the Mercator projection used for a sphere and, instead, employing a conical projection as the texture source (that is, a circle whose circumference corresponds to the base of the cone and whose center corresponds to the tip). The discontinuities in shading, however, cannot be overcome with triangles or quadsplits.
The cone example revealed a bug in the pixel shader (Listing 2). In
subroutine BarycentricWeights, the following assignment fails if A is
zero:
t[i] = (r[i]*r[(i+1)%4]D)/A;
It should be replaced with:
t[i] = abs(A) < 1.e6? 0 : (r[i]*r[(i+1)%4]D)/A;
This correction removes the need for double precision in the subroutines
BarycentricWeights and UV (thus, p. 73, “Double precision is used ...
artifacts” is incorrect).