We only consider surfaces that are topologically equivalent to a sphere, that is, surfaces that can be smoothly deformed into a spherical shape. Note that when is a superscript of a spherical harmonic or associated Legendre function, as in or , it identifies a unique function and is not an exponent.
Spherical harmonics are complex valued functions.
Real valued functions are obtained by forming appropriate linear combinations of the complex functions. Spherical harmonics are ideal for representing functions defined on a unit sphere since they form a complete orthonormal basis set. Approximations are computed to a finite value of which is called the order of the expansion. This approximation procedure has been applied to molecular surfaces by Max and Getzoff, and by Leicester et. In both of these methods, the function defines the radius function , that is, the distance from an origin inside the surface to the surface point with angular coordinate .