This page was last edited on 5 Mayat However, this text is one of my Ph. Spherical harmonics If you are a seller for this product, would you like to suggest updates through seller support? AmazonGlobal Ship Hsortley Internationally. In 3D computer graphicsspherical harmonics play a role in a wide variety of topics including indirect lighting ambient occlusionglobal illuminationprecomputed radiance transferetc.
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The spherical coordinates used in this article are consistent with those used by physicists, but differ from those employed by mathematicians see spherical coordinates. The terms in the first summation approach zero as r goes to infinity, whereas the terms in the second summation approach zero at the origin. Orthogonality and normalization Several different normalizations are in common use for the spherical harmonic functions.
Alternatively, this equation follows from the relation of the spherical harmonic functions with the Wigner D-matrix. Condon-Shortley phase One source of confusion with the definition of the spherical harmonic functions concerns a phase factor of -1 m, commonly referred to as the Condon-Shortley phase in the quantum mechanical literature. In the quantum mechanics community, it is common practice to either include this phase factor in the definition of the associated Legendre functions , or to append it to the definition of the spherical harmonic functions.
There is no requirement to use the Condon-Shortley phase in the definition of the spherical harmonic functions, but including it can simplify some quantum mechanical operations, especially the application of raising and lowering operators. The geodesy and magnetics communities never include the Condon-Shortley phase factor in their definitions of the spherical harmonic functions.
Spherical harmonics expansion The spherical harmonics form a complete set of orthonormal functions and thus form a vector space analogous to unit basis vectors. On the unit sphere, any square-integrable function can thus be expanded as a linear combination of these: This expansion is exact as long as goes to infinity. Truncation errors will arise when limiting the sum over to a finite bandwidth L.
The expansion coefficients can be obtained by multiplying the above equation by the complex conjugate of a spherical harmonic, integrating over the solid angle , and utilizing the above orthogonality relationships. For the case of orthonormalized harmonics, this gives: An alternative set of spherical harmonics for real functions may be obtained by taking the set: where N l,m denotes the normalization constant as a function of l and m. These functions have the same normalization properties as the complex ones above.
Note, however, that the listed functions differ by the phase -1 m from the phase given in this article. Spectrum analysis The total power of a function f is defined in the signal processing literature as the integral of the function squared, divided by the area it spans. In a similar manner, one can define the cross-power of two functions as where is defined as the cross-power spectrum. If the functions f and g have a zero mean i.
It is common that the cross- power spectrum is well approximated by a power law of the form Addition theorem A mathematical result of considerable interest and use is called the addition theorem for spherical harmonics.
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