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- get understanding of ordinary vector linear algebra.
- understand what vector dot product does and why
- understand why an orthogonal set of basis vectors for the space you're working in is useful / what properties it has / how its used. like basic euclidean 3d space (1,0,0) (0,1,0) (0, 0, 1) basis vectors.
- get a refresher on basic calculus, in particular integrals
- understand this inner product, it's a generalization of dot product, except you can think of your vectors having infinite number of dimensions now.
- the properties of the dot product you know (like that two vectors are perpendicular if their dot product is 0) work for the inner product too. or perhaps its better to say that the general inner product is defined to have similar properties
- there are functions that are orthogonal to each other in the same way vectors can be orthogonal to each other, and you can use the inner product to tell which ones.
- spherical harmonics are constructed / by design orthogonal to each other. how to show this and where the intuition for finding them could come from is a whole topic...
- but once you have it, just like you can project vectors onto basis vectors (to essentially transform them into the coordinate system described by those basis vectors), you can project functions into the coordinate system represented by those orthogonal functions.
- then you have to figure out why you would even want to do this. in short is has a lot of useful properties/applications. in the graphics case you can compress some quite complex functions into just a few coefficients using this (not perfectly, there is some 'information loss', but still). integrating over two functions becomes cheaper when they are projected to SH basis. it lets you do some unintuitive stuff like combine light that goes into different directions into one common set of coefficients.
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.
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 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.
https://en.wikipedia.org/wiki/Ambisonics#Higher-order_ambiso...
https://en.wikipedia.org/wiki/Atomic_orbital#Orbitals_table
...and the same patterns appear on the unit disk with the Zernike polynomials, used to describe optical aberrations and more.
https://en.wikipedia.org/wiki/Zernike_polynomials
Or it can give game audio to one end?
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?
What’s interesting is if the environment is not spherically symmetric (consider an electron in a molecule) the solutions to the wave equation (the electronic wave functions) are no longer spherical harmonics, even though we like to approximate them with combinations of spherical harmonic basis functions centered on each nucleus. It’s kind of like standing waves on a circular drum head (hydrogen atom) vs standing waves on an irregular shaped drum head
Of course the nucleus also has a wave nature and in reality this interacts with the electrons, but in chemistry and materials we mostly ignore this and approximate the nucleus like a static point charge from the elctrons perspective because the electrons are so much lighter and faster
In other words you can express any reasonable function on the unit sphere as a series of spherical harmonic terms. That makes them ideal for working with differential equations (eg schrodinger's equation for the hydrogen atom, or, emission from an arbitrary light source).
I guess at a certain point the number of terms becomes so large that it makes sense to just use a cube map?
Like, if you know the third order harmonics that's only 16 values you have to pass around