The purpose of this package is to provide methods to numerically handle real spherical harmonics expansions in Cartesian coordinates.
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 .
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 .
can be transformed from from spherical to Cartesian coordinates, where is can be expressed as a homogeneous polynomial of degree .
MultivariatePolynomials.Polynomial representation of
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
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
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[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
SphericalHarmonicCoefficients(C) will throw an error if
length(C) is not for some . From there on the corresponding polynomial representation in cartesian coordinates
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
If we change from a coordinate sytsem with coordinates
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) sphericalHarmonicsExpansion(cTranslated,u,v,w) 2.1850968611841584uw - 1.0925484305920792vw + 2.1850968611841584u - 1.0925484305920792v + 11.847981254502878
If you want to evaluate at a specific point you can use the standard interface of
f(x=>0.5, y=>-1.0, z=>0.25) 12.394255469798921 f((x,y,z)=>(0.5,-1.0,0.25)) 12.394255469798921
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 g(0.5,-1.0,0.25) 12.394255469798921
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) h(0.5,-1.0,0.25) 12.394255469798921
GeneralizedGenerated for function generation and comes with a significant overhead.
For more informations on the
MultivariatePolynomials package please visit the project page on github.
19 days ago