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The table of spherical harmonics contains a list of common spherical harmonics. Since the spherical harmonics form a complete set of orthogonal functions and thus an orthonormal basis , every function defined on the surface of a sphere can be written as a sum of these spherical harmonics.
This is a table of orthonormalized spherical harmonics that employ the Condon-Shortley phase up to degree =. Some of these formulas are expressed in terms of the Cartesian expansion of the spherical harmonics into polynomials in x , y , z , and r .
The colatitude angle in spherical coordinates is the angle used above. The longitude angle, , appears in a multiplying factor. Together, they make a set of functions called spherical harmonics. These functions express the symmetry of the two-sphere under the action of the Lie group SO(3). [citation needed]
Under rotation, vector spherical harmonics are transformed through each other in the same way as the corresponding scalar spherical functions, which are generating for a specific type of vector harmonics. For example, if the generating functions are the usual spherical harmonics, then the vector harmonics will also be transformed through the ...
The zonal spherical harmonics are rotationally invariant, meaning that () = () for every orthogonal transformation R.Conversely, any function f(x,y) on S n−1 ×S n−1 that is a spherical harmonic in y for each fixed x, and that satisfies this invariance property, is a constant multiple of the degree ℓ zonal harmonic.
Associated Legendre polynomials are the colatitudinal part of the spherical harmonics which are common to all separations of Laplace's equation in spherical polar coordinates. [2] The radial part of the solution varies from one potential to another, but the harmonics are always the same and are a consequence of spherical symmetry.
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Spherical harmonics are often used to approximate the shape of the geoid. The current best such set of spherical harmonic coefficients is EGM2020 (Earth Gravitational Model 2020), determined in an international collaborative project led by the National Imagery and Mapping Agency (now the National Geospatial-Intelligence Agency , or NGA).