# A Lie Theoretic Proposal on Algorithms for the Spherical Harmonic Lighting

## Abstract

The spherical harmonics are the angular portion of the solution to the Laplace equation in spherical coordinates and provide a frequency-basis for representing functions on the sphere. The spherical harmonic lighting, as defined by Robin Green at Sony Computer Entertainment in 2003, is a family of real-time rendering techniques that may produce certain realistic shading and shadowing with relatively small overhead lighting. All such spherical harmonic lighting techniques involve replacing parts of standard lighting equations with spherical functions that have been projected into a frequency space using the spherical harmonics as a basis (or a weight space of irreducible finite dimensional representation of the rotation group). In this chapter, using a group theoretical background of spherical harmonics and rather simple realization of the space of functions on the two dimensional sphere in the frame work of representation theory, we propose a possible geometry preserving algebraic/efficient computing, which might accelerate the (numerical and exact) computations slightly for spherical harmonic lighting.

### Keywords

Spherical harmonic lighting Global illumination Spherical harmonics Irreducible representation Intertwiner Casimir element Legendre polynomials### References

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