# Tag Info

10

You may find the expansion of a plane wave in spherical waves to be helpful here: $$e^{i\mathbf{k}\cdot\mathbf{x}} = 4\pi\sum_{l=0}^\infty\sum_{m=-l}^l i^lj_l(kr)Y_{lm}(\theta,\phi)Y_{lm}^*(\vartheta,\varphi)$$ where $\theta$, $\phi$ are the angular variables for $\mathbf{x}$ and $\vartheta$, $\varphi$ for $\mathbf{k}$; the radial functions $j_l$ are the ...

5

Overview These types of polynomials are used in quantum chemistry, potential theory, magnet shimming, and probably many other branches of science. One problem is that the nomenclature seems to be subtly different between the fields, so the following description is not definitive. Let us call the solutions of the Laplace equation $\Delta\Phi = 0$, when they ...

3

Given samples of a function $f(\theta,\phi)$, you will need to numerically evaluate the integral $$\int_0^\pi\int_0^{2\pi}f(\theta,\phi)\left[Y_n^{m}(\theta,\phi)\right]^*\sin\theta d\phi d\theta\, ,$$ to obtain the $n,m$th coefficient of your expansion. Hopefully your samples are at some convenient locations on the unit sphere, either uniformly sampled or ...

2

After some more searching I've managed to find what I was looking for. They are available on NETLIB as stripack and ssrfpack - fortran routines - that allow for spherical interpolation of irregular data using spherical splines. There's also an alternative that uses generalized Green's function for spherical surface splines in tension. The matlab code (and ...

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If I'm not mistaken, these spin-weighted spherical harmonic functions are equivalent to the Generalized Associated Legendre functions. In the reference work Virchenko & Fedotova, you can find recurrence relations in chapter 5, p32. Chapter 15, p. 96, of the same reference discusses integral transforms with the Generalized Associated Legendre functions.

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Deriving the CFL condition was handled by @nicoguaro in the comments. The coordinate singularity problem at $r = 0$ was addressed in this paper: "Numerical Treatment of Polar Coordinate Singularities" Mohseni + Colonius JCP (2000) and is available here. From the introduction: In the present paper we investigate a method for treating the coordinate ...

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To do precisely what you are describing, you likely want to interpolate your grid data to a set of discrete spheres with prescribed radii, then perform the spherical transform on each set of data. Another possibility, and a generalization of the above, would be to define a spherical basis set with the spherical harmonics as the $(\theta, \phi)$-varying ...

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On a basic search on Spherical Interpolation I found these: Graphics Math Template Library (GMTL) Link: http://ggt.sourceforge.net/ How do I perform spherical interpolation with quaternions? Use the slerp function. You need an origin quaternion, a target quaternion, and an interpolation amount between 0 and 1. The following example interpolated ...

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