Stratified Sampling of Convex Quadrilaterals

Abstract

We derive a closed-form expression for a smooth uniform bijection from a unit square onto an arbitrary convex planar quadrilateral; that is, a smooth parameterization of the quadrilateral under which the image of equal areas remain of equal area. The properties of the mapping make it well-suited to stratified Monte Carlo sampling and therefore useful for illumination computations. We begin with a simple bilinear mapping from the unit square onto the quadrilateral, then derive a warping function, from the unit square to itself, which results in a uniform map onto the quadrilateral when composed with the original bilinear map. The resulting sampling algorithm requires only a few lines of code with no iteration or branching.