On Sat, Jul 09, 2011 at 09:41:04PM -0400, Marc Lavallée wrote: > If you could help me understand spherical harmonics, I'd be a "MAG > fanboy" in no time. The best didactic resource I found is a very > strange article titled "Notes on Basic Ideas of Spherical Harmonics". > It's so good that I barely understand 10% of it.
What you probably need is some intuitive understanding of them. I've tried it many times, and the following seems to work well with most people interested in the subject. You are probably familiar with the fact that a cyclic waveform, e.g. a square wave, corresponds to an harmonic line spectrum: if the fundamental frequency is F, the waveform is the sum of a number of sine/cosine waves with frequencies k * F, with k an integer. The thing that connects the two representations, the waveform as a function of time and the spectrum, is the Fourier trans- form or its inverse. We can switch between the two at any time without loss of information, both one cycle of the waveform and its spectrum contain all there is to know about the waveform. Visualise the waveform as a function of time, with time on the x-axis, and cut out a piece corresponding to one cycle. We can bend this piece of x-axis into a circle. Now instead if interpreting that axis (now a circle) as 'time' we can interpret it as an angle: every point on the circle corres- ponds to a direction (as seen from the center). So anything that is a function of e.g. direction in the horizontal plane can be represented by a 'spectrum' as well. Can we generalise this to directions not just in the H plane but in 3-D space ? Let's try using a 2-D Fourier transform, just as we used a 1-D FT for 2-D space (a plane). The equivalent of a cyclic function in that case is plane consisting of identical square tiles - it is cyclic both in x and y, and the 2-D Fourier transform can be used to compute its spectrum (a very common thing e.g. in image processing). We can cut out one square, just as we did with the single period before, and try to bend it into a sphere since directions in 3-D space correspond to points on a sphere. We can take the top and bottom edges and bring them together, forming a tube. Now we can bend the tube to bring its two ends together. In both cases the points that meet have the same function value, so we preserve the cyclic nature of our function. But the result is not sphere as we would want, but a torus. Can we bend somehow our square into a sphere and such that identical points on the edges are brought together ? The answer is no, it can't be done. For both a torus and a sphere we can identify any point on it with two coordinates, e.g. azimuth and elevation, but they are fundamentally different surfaces. On a torus the two coordinates are really independent, on a sphere they are not. So we no know that a 2-D Fourier transform can't be used to find the spectrum of a function defined on the sphere, as we could do using the 1-D FT on a circle. Then _what_ does correspond to the components of a spectrum on a sphere ? This turns out to be the set of functions called Spherical Harmonics. They arise quite naturally when trying to solve some equations (e.g. the wave equation) in 3-D space using spherical coordinates instead of x,y,z, just as sine and cosine appear as the solutions of similar but simpler equations. Ciao, -- FA _______________________________________________ Sursound mailing list [email protected] https://mail.music.vt.edu/mailman/listinfo/sursound
