Spherical Harmonics: Difference between revisions
| (3 intermediate revisions by the same user not shown) | |||
| Line 1: | Line 1: | ||
Spherical Harmonics are a set of orthonormal basis functions defined over a sphere.<br> | Spherical Harmonics are a set of orthonormal basis functions defined over a sphere.<br> | ||
<math>f: (\phi, \theta) \ | <math>f: (\phi, \theta) \mapsto f(\phi, \theta) \in \mathbb{R}</math> | ||
==Background== | ==Background== | ||
| Line 16: | Line 16: | ||
\(P^0_0(x)\) | \(P^0_0(x)\) | ||
\(P^0_1(x), P^1_1(x)\) | \(P^0_1(x), P^1_1(x)\) | ||
\(P^0_2(x), P^1_2(x), | \(P^0_2(x), P^1_2(x), P^2_2(x)\) | ||
The following 3 recurrance relations define the associated legendre polynomials: | The following 3 recurrance relations define the associated legendre polynomials: | ||
| Line 34: | Line 34: | ||
==Definition== | ==Definition== | ||
Spherical Harmonics are a set of orthonormal basis functions defined on the sphere. | Spherical Harmonics are a set of orthonormal basis functions defined on the sphere. | ||
Below are some explicit formulas for Laplace spherical harmonics stolen from Sloan<ref name="stupidsh">Peter-Pike Sloan, [http://www.ppsloan.org/publications/StupidSH36.pdf Stupid Spherical Harmonics (SH) Tricks]</ref>. | Below are some explicit formulas for Laplace spherical harmonics stolen from Sloan<ref name="stupidsh">Peter-Pike Sloan, [http://www.ppsloan.org/publications/StupidSH36.pdf Stupid Spherical Harmonics (SH) Tricks]</ref>. You can also find alternative equations in DLMF<ref name="dlmf">Digital Library of Mathematical Functions, 14.30. [https://dlmf.nist.gov/14.30 https://dlmf.nist.gov/14.30]</ref>. | ||
There are <math>2l+1</math> functions for each band. | There are <math>2l+1</math> functions for each band. | ||
| Line 73: | Line 74: | ||
</math> | </math> | ||
}} | }} | ||
==Operations== | |||
===Addition=== | |||
Just add the coefficients | |||
===Multiplication=== | |||
===Rotation=== | |||
==Visualizations== | ==Visualizations== | ||