In case the author is reading this, if you're going to introduce the complex-valued harmonics you should be careful to put the complex conjugate in the inner product
<f, g> = ∫ f(ω)^* g(ω) dω
which does match the corresponding linear-algebra inner product if the vectors are over the complex numbers
p . q = Σ_i p^*_i q_i
which guarantees that p.p ≥ 0 even for complex p (and does not change the only-real case).
what level of math do I need to understand this? or the rest of the math in the post, something I can catch up on in a weekend? barely remember the last math class I took seriously, trig like 18 years ago
- 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.
> what level of math do I need to understand this?
A basic understanding of differential equations is all that's really necessary, but knowing about orthogonal polynomials would be helpful too.
> something I can catch up on in a weekend?
Probably not. If you know any calculus (even a basic high school class should be enough), two weekends would probably be enough; if you don't know calculus, then double it.
My advice would be to use an introductory level quantum physics textbook or an advanced chemistry textbook, since the spherical harmonics are used quite a bit in those fields. You could use a math textbook too, but those will tend to focus on details that are irrelevant to you.
An alternate path would be to learn about Fourier series/transformations, then what's discussed in the article will follow as a natural consequence. This is probably a harder option, but there's lots of really good learning materials online for Fourier transformations (and comparatively little for spherical harmonics), so it may end up being easier for you.
