A Julia package to handle spherical harmonic functions



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The purpose of this package is to provide methods to numerically handle real spherical harmonics expansions in Cartesian coordinates.

Table of Contents

Mathematical Background

Definition of the Spherical Harmonics

The normalized real spherical harmonics on the unit sphere are defined by

where and , and are the spherical angular coordinates,

is the normalization factor and

are the associated Legendre polynomials which can be derived from the Legendre polynomials

Note that you will also find a convention in literature, where the are scaled by .

Spherical Harmonics Expansions

Each function satisfying Laplace's equation in a region can be written as a spherical harmonic expansion

for all , where denote the spherical coefficients and .

The term

can be transformed from from spherical to Cartesian coordinates, where is can be expressed as a homogeneous polynomial of degree .


Polynomial Representation of the Spherical Harmonics

Generate a MultivariatePolynomials.Polynomial representation of

in variables α, β, and γ on the unit sphere by

using SphericalHarmonics
@polyvar α β γ
l = 7 
m = -2

p = ylm(l,m,α,β,γ)
63.28217501963252αβγ⁵ - 48.67859616894809αβγ³ + 6.63799038667474αβγ

The polynomial representation of

in variables x, y, and z on can be obtained by

@polyvar x y z

p = rlylm(l,m,x,y,z)
6.63799038667474x⁵yz + 13.27598077334948x³y³z - 35.40261539559861x³yz³ + 6.63799038667474xy⁵z - 35.40261539559861xy³z³ + 21.24156923735917xyz⁵

Polynomial Representation of the Spherical Harmonics Expansions

In case where a function is equal to or can be approximated by a finite Spherical harmonic expansion

with its multivariate polynomial representation has finite degree.

Coefficents can be initialized and populated by by c[l,m] = 42.0.

L = 2
c = SphericalHarmonicCoefficients(L)
c[0,0] = 42.0 #c₀₀
c[2,-1] = -1.0 #c₂₋₁
c[2,1] = 2.0 #c₂₁

Internally the coefficients are lexicographically stored in a vector (c[0,0], c[1,-1], c[1,0], c[1,1], c[2,-2], ...). So the above initialization is equivalent to

C = [42.0,0,0,0,0,-1,0,2,0]
c = SphericalHarmonicCoefficients(C)
f = sphericalHarmonicsExpansion(c,x,y,z)
2.1850968611841584xz + -1.0925484305920792yz + 11.847981254502882

Note that SphericalHarmonicCoefficients(C) will throw an error if length(C) is not for some . From there on the corresponding polynomial representation in cartesian coordinates x, y, and z can be obtained by

@polyvar x y z

f = sphericalHarmonicsExpansion(c,x,y,z)
2.1850968611841584xz - 1.0925484305920792yz + 11.847981254502882

Currently, expansions up to $L=66$ are supported

Transformation of Expansion Coefficients under Translation

If we change from a coordinate sytsem with coordinates x, y, and z into a translated one with new coordinates u = x + tx, v = y + ty, and w = z + tz we need transformed coefficients to express the expansiion in these new coordinates. To this end we can do

@polyvar u v w
translationVector = [0,0,1.0] # [tx,ty,tz]

cTranslated = translation(c,translationVector)
2.1850968611841584uw - 1.0925484305920792vw + 2.1850968611841584u - 1.0925484305920792v + 11.847981254502878

Numerical Evaluation

If you want to evaluate at a specific point you can use the standard interface of MultivariatePolynomials

f(x=>0.5, y=>-1.0, z=>0.25)

In case where you want to evaluate for a large number of points you might run into performance issues. To this end we provide two methods to dynamically generate fast evaluating functions. Either use

g = @fastfunc f

which has moderate generation overhead. Usage from within local scope requires Base.invokelatest(foo, 1.0,2.0,3.0) instead of foo(1.0,2.0,3.0) to avoid issue #4. Or use

h = fastfunc(f)

which uses GeneralizedGenerated for function generation and comes with a significant overhead.

Further Reading

For more informations on the MultivariatePolynomials package please visit the project page on github.

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