Spherical Harmonics
Solving for Spherical Harmonic Analysis Quadrature Weights
In “More Notes on Calculating the Spherical Harmonics”, I used a simple iterative approach to converge on relatively accurate analysis of coefficients from a pixelized image on the sphere. In this article, we will look at the other method to improve the accuracy of approximating integration with finite sums by including so-called quadrature weight factors.
Spherical Harmonic Transforms on Ring-based Pixelizations
In two previous articles, I described the details of performing spherical harmonic synthesis and analysis using an equidistant cylindrical projection (ECP) pixelization scheme as the assumed map format. In this article, I’ll describe the generalizations to the algorithms which permit supporting the transform of any “ring-based” pixelization of the sphere.
More Notes on Calculating the Spherical Harmonics
This article is a long-overdue follow up to Notes on Calculating the Spherical Harmonics which considers the analysis of real-space map on the sphere to its spherical harmonic coefficients.
Plots of the Spherical Harmonics Eigenmodes
This posting is a simple addendum to the much longer article, Notes on Calculating the Spherical Harmonics. Because it had already grown so long, I left out making any plots of individual eigenmodes, so here I simply present plots of the first 50 degrees of the pure spherical harmonic eigenmodes, which can be useful to help visualize some of the symmetry properties that were discussed in the longer article.
Notes on Calculating the Spherical Harmonics
In this article I review the critical properties of the Spherical Harmonics. In particular, I concentrate on filling in a couple of details regarding numerical computation of the spherical harmonic transformations that I have found to be unstated or less clear than ideal in the literature.
Pre-normalizing Legendre Polynomials
In the previous part we computed Legendre polynomials via a few recurrence relations, but the polynomial values grow rapidly with degree and quickly overflow the range of standard finite floating point numbers. In this posting we explore baking in a normalization factor into the recurrence relation — such as used when the Legendre polynomials are used to calculate the spherical harmonics — that eliminates the overflow.