Spherical harmonics appear in several computer graphics techniques. I feel that in order to be a better computer graphics developer, I need to have a deep understanding of what they are and how they are used.

It seems that the reference most often recommended to understand Spherical Harmonics is "Stupid Spherical Harmonics Tricks" by Peter-Pike Sloan.

I started reading it but did not find a "satisfying" definition of SH, it seems like the document mostly relies on other references for the "basics". Other references introduce the Fourier Basis functions as a "simpler version" of SH, but once again it seems hard to find good material explaining them.

What are thorough, accessible references to understand Fourier basis functions and Spherical Harmonics ?

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    $\begingroup$ What is missing for you from the SH wikipedia page ? en.wikipedia.org/wiki/Spherical_harmonics $\endgroup$ Commented Oct 15, 2015 at 13:20
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    $\begingroup$ Fourier basis functions are simply sines (or sines.x * sines.y). Like sound can decompose in spectrums (i.e. set of sines), so is it for a 1D drawing. For 2D data, you simply do that in x then y (but I find quite intuitive to extrapolate the notion of wavelength to 2D). Maybe you should tell what is your background in maths ? (which school level ?) $\endgroup$ Commented Oct 15, 2015 at 13:23
  • $\begingroup$ @FabriceNEYRET Thank you for the link. While I don't think this information will benefit other users, my background in maths is prépa(MP*) + engineering school (Master Degree) ; the program did not have in-depth covering of SH. I am indeed looking for material as rigorous as what I was used to study, although I now have less time to do so, so brevity is also important. $\endgroup$
    – wip
    Commented Oct 16, 2015 at 3:02
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    $\begingroup$ I read the paper you linked, it seems pretty solid. $\endgroup$
    – joojaa
    Commented Oct 16, 2015 at 5:10

1 Answer 1


wil, you largely have enough scholar background, you must have done Fourier and Laplace transforms in second year, and maybe in your engineering school again as part of signal processing classes.

If you read "stupid tricks" there is not much more you can do to find a condensed course at this point.

The second most famous paper that goes with SH for graphics is by Robin Green called "the gritty details":

And the third most important is the one by Ramamoorthi (the original paper preceding "An Efficient Representation for Irradiance Environment Maps"), which was called On the relationship between radiance and irradiance: determining the illumination from images of a convex Lambertian object

And I think they mention somewhere that SH were previously most used by another science field, forgot which one, physics maybe, and that most of their base material came from these papers. So if you want to dig in the roots, you've got to pull out these mid last century references.

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    $\begingroup$ Indeed the way Fourier is treated at university or enginneering school can be totally different from place to place, from a pure fancy pure math toy in strange mathematical spaces (but here it's even worse for Finite Element methods) to something really connected to signal, theory and applications. Similarily SH are sometime presented as cool math object dedicated to the physics of electronic orbitals. In all these case it's not easy for (ex)students to transpose to CG or signal. -> that's why it's important to tell where he starts from and where he seeks to. $\endgroup$ Commented Oct 16, 2015 at 8:06

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