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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.
Below the complex spherical harmonics are represented on 2D plots with the azimuthal angle, , on the horizontal axis and the polar angle, , on the vertical axis.The saturation of the color at any point represents the magnitude of the spherical harmonic and the hue represents the phase.
For example, one sphere that is described in Cartesian coordinates with the equation x 2 + y 2 + z 2 = c 2 can be described in spherical coordinates by the simple equation r = c. (In this system— shown here in the mathematics convention —the sphere is adapted as a unit sphere , where the radius is set to unity and then can generally be ...
Finite spherical symmetry groups are also called point groups in three dimensions. There are five fundamental symmetry classes which have triangular fundamental domains: dihedral , cyclic , tetrahedral , octahedral , and icosahedral symmetry.
An important geometry related to that of the sphere is that of the real projective plane; it is obtained by identifying antipodal points (pairs of opposite points) on the sphere. Locally, the projective plane has all the properties of spherical geometry, but it has different global properties.
Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are great circles. Spherical trigonometry is of great importance for calculations in astronomy, geodesy, and ...
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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 ...