Constant Time Parallel Indexing of Points in a Triangle

Consider a triangle whose three vertices are grid points. Let k denote the number of grid points in the triangle. We describe an indexing of the triangle: a bijective mapping from {0, ..., k-1} to the grid points in the triangle. Computing such a mapping is a fundamental subroutine in fine-grained parallel computation arising in graphics applications such as ray-tracing. We describe a very fast indexing algorithm: after a preprocessing phase requiring time proportional to the number of bits in the vertices of the triangle, a grid point in the triangle can be computed in constant time from its index. The method requires only constant space.