Introduction to spherical harmonics for graphics programmers

+1 for introducing them as real-valued functions over cartesian coordinates!

Typically, spherical harmonics are introduced as a complex function over spherical coordinates, which makes them much easier to derive, but imo hides their beauty.

The real-valued, cartesian form of regular spherical harmonics is also called "solid harmonics" or "harmonic polynomials", in case you want to dig deeper.

If one needs to describe (and maybe compress) functions or data on a sphere, spherical harmonics are really a thing.

An alternative would be to construct a new function (or matrix) that is not only periodic in azimuth, but also in elevation (i.e., extend elevation to a full circle -pi to +pi). Then, one can simply compute two independent Fourie r transforms: along azimuth and along elevation. [1] The same idea works on matrices using the Discrete Fourier transform (DFT/FFT). However, you then have to accept things like that your data points are all equal at the poles.

[1] https://en.wikipedia.org/wiki/Double_Fourier_sphere_method

This is really great. I always saw those harmonic shapes as electron orbitals, I had no idea they could be used in lighting too - so cool.

It made me wonder - why do the electron orbitals take those shapes in say a hydrogen atom? Is there a constraint on the electron and proton together that make it fit only to spherical harmonic functions?

> spherical harmonics can have uses beyond lighting

This math is also used in Ambisonic surround sound though newer techniques use planewave expansion.

For games, the full-sphere encoding of Ambisonic B-format can be decoded for arbitrary speaker locations and the soundfield rotated around any axis. I'm not sure if its ever been used for a game though.

Awesome write up. Is the appeal of spherical harmonics in graphics compression?

Like, if you know the third order harmonics that's only 16 values you have to pass around

Author here. Please let me know if the sample code doesn't work for you. It's all single threaded dumb JavaScript which makes it very easy to read, but definitely not performant. I decided to stick with it for didactic reasons, but still worried that it may hang someone's browser.