Eigenfunction expansions of the Laplace-Beltrami Operator

In mathematics, spherical harmonics are the angular portion of a set of solutions to Laplace's equation. Spherical harmonics are important in many theoretical and practical applications, particularly in the computation of atomic orbital electron configurations, representation of gravitational fields, geoids, and the magnetic fields of planetary bodies and stars, and characterization of the cosmic microwave background radiation. In 3D computer graphics, spherical harmonics play a special role in a wide variety of topics including indirect lighting (ambient occlusion, global illumination, precomputed radiance transfer, etc.) and recognition of 3D shapes. In this chapter we give brief introduction to theory of spherical harmonics on the sphere.