Has anyone seen this effect before? I made polar UV coordinates for the ring and applied the standard brick texture. The polar coordinates basically take a square in UV space and squash it onto a trapezoidal face. I would have expected linear interpolation throughout the face, but something odd happens near the middle. In the top left quadrant, I used ctrl-r to split the faces and line up the split edges with the texture lines. That straightens them out, so I'm sure the linear calculation is correct along all the face edges.
After taking the picture, I tried splitting a distorted quad along the diagonal, switching between fixed and fixed-alternate. It turns out, one of the split directions matches the distortion, and the other direction distorts the other way. So behind the scenes, quads are made into triangles for the UV mapping.
Unless there is a different UV calculation mode for blender, I think the only fix is to split the trapezoids radially.