Sphere Maps with the Near-Equal Solid-Angle Property

Liang Wan, and Tien-Tsin Wong
Game Developers Conference 2007

Summary

Unlike the common spherical maps that sample the sphere unevenly, three sphere maps that own the near-equal solid-angle property are introduced. The first is HEALPix, adopted from astrophysics, which partitions the sphere into equal-area quadrilateral elements. It allows fast integration over the sphere and is especially useful in environment sampling. The second is one of our newly designed maps. It subdivides the sphere with a set of simple equations. Unlike HEALPix, it fits naturally into the six-face cube structure and can fully utilize the hardware operators tailored for traditional cubemap. It samples the sphere in a more balanced fashion than cubemap. The third one uses a family of skew great circles as the subdivision strategy. All texels are distorted similarly. Besides the mathematics, we present the real-time demos to demonstrate their applicabilities to real-time application.

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Visual Comparison

The table below compares the properties of several spherical maps.
　

 

Cubemap

HEALPix

 Rhombic
Dodecahedron

Isocube

Base face
number

  6

  12

  12

  6

Uniform
distribution

  Uneven

 Even

Fair

Even

Equal-area
property

Not Equal

Equal

Near

Equal

Texel
shape

Quad

Quad

Quad

Quad/Tri

Base

Polyhedron

Spherical
partitioning

Polyhedron

Spherical
partitioning

Related Publications

Isocube:

  • "Isocube: Exploiting the Cubemap Hardware", L. Wan, T. T. Wong and C. S. Leung, IEEE Transactions on Visualization and Computer Graphics, Vol. 13, No. 4, pp. 720-731, 2007.

　

Rhombic:

  • Chi-Wing Fu, Liang Wan, Tien-Tsin Wong, and Chi-Sing Leung, The Rhombic Dodecahedron Map: An Efficient Scheme for Encoding Panoramic Video, IEEE Transactions on Multimedia, Vol. 11, No. 4, pp. 634-644, December 2009.

　

HEALPix:

  • "Real-Time Environment Mapping with Equal Solid-Angle Spherical Quad-Map", T. T. Wong, L. Wan, C. S. Leung and P. M. Lam, Shader X4: Advanced Rendering Techniques, Edited by W. Engel, Charles River Media, 2006, pp. 221-233.

　

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