Optimal sampling for hemicubes

The hemicube estimates of form factors are based on a finite set of sample directions. We obtain several optimal arrangements of sample directions, which minimize the variance of these estimates. They are based on changing the size or shape of the pixels or the shape of the hemicube, or using non-uniform pixel grids. The best reduces the variance by 43%. The variance calculation is based on the assumption that the errors in the estimate are caused by the projections of single polygon edges, and that the positions and orientations of these edges are random. This replaces the infinite dimensional space of possible environments by the two dimensional space of great circles on the unit sphere, making the numerical variance minimization possible