Area-Preserving Parameterizations for Spherical Ellipses

Ibón Guillén, Carlos Ureña, Alan King, Marcos Fajardo, Iliyan Georgiev, Jorge López-Moreno, Adrian Jarabo
EGSR 2017

Our mappings enable the stratified sampling of the solid angle subtended by oriented disk light sources (left), which can yield substantially lower error than traditional uniform area sampling (right). With 256 samples/pixel our tabulated radial map yields a noise-free image, while area sampling suffers from a significant amount of noise. This scene, rendered in Arnold, showcases many production features, including high-resolution texture maps, fur, displacement, subsurface scattering, indirect surface and volume-to-surface light transport. In such scenes the higher cost of our method has a negligible impact on the total rendering performance.


We present new methods for uniformly sampling the solid angle subtended by a disk. To achieve this, we devise two novel area-preserving mappings from the unit square to a spherical ellipse (i.e. the projection of the disk onto the unit sphere). These mappings allow for low-variance stratified sampling of direct illumination from disk-shaped light sources. We discuss how to efficiently incorporate our methods into a production renderer and demonstrate the quality of our maps, showing significantly lower variance than previous work.

Downloads and links

  • paper (PDF, 5.1 MB)
  • supplemental - numerical validation and additional results (PDF, 3.6 MB)
  • slides - from the conference presentation (PDF, 6.2 MB)
  • citation (BIB)

BibTeX reference

    Author = {Guill\'{e}n, Ib\'{o}n and Ure\~{n}a, Carlos and King, Alan and Fajardo, Marcos and Georgiev, Iliyan and L\'{o}pez-Moreno, Jorge and Jarabo, Adrian},
    Journal = {Computer Graphics Forum (Proceedings of EGSR)},
    Number = {4},
    Title = {Area-Preserving Parameterizations for Spherical Ellipses},
    Volume = {36},
    Year = {2017}