Stratified Samlping of Convex Quadrilaterals

James Arvo
University of California, Irvine

Kevin Novins
University of Auckland

This paper appears in issue Volume 12, Number 2.
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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.

Author Information

James Arvo, Information and Computer Science, 430E Computer Science Building, Irvine, CA 92697-3425 arvo@uci.edu

Kevin Novins, Department of Computer Science, Private Bag 92019, Auckland 1001, New Zealand novins@cs.auckland.ac.nz

BibTeX Entry

@article{ArvoNovins07,
  author = "James Arvo and Kevin Novins",
  title = "Stratified Samlping of Convex Quadrilaterals",
  journal = "journal of graphics tools",
  volume = "12",
  number = "2",
  pages = "1-12",
  year = "2007",
}