08-06-2005 03:08									#1
	Rotation of the spherical harmonics Support Apolyton buy from Amazon

Any good references out there?

Basically, my problem is that I have the coefficients of the spherical harmonic decompositon of a function defined on the sphere. The only way I was able to write these down was to define my z axis such that the function was properly azimuthally symmetric.

My decomposition therefore has Clm = 0 if m is not 0.

What I want to be able to do is transform this decomposition into another basis with new z axis pointing in arbitrary theta, phi direction

What I remember of this is that the transformation involves only a sum over m's (not l's) therefore it shouldn't take too much computer time to do (which is good, because I have to transform the bases approximately 3 million times, before doing some other manipulations)

this has something to do with the wigner-eckart theorem, but I don't want to think too hard about it. Therefore, the more cookbooky the reference gets the better.

	08-06-2005 03:11									#2
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I'm already getting way too clever for my own good right now.

I used Rodrigues' formula today. That wasn't something I expected to do.

	08-06-2005 03:33									#4
	Re: Rotation of the spherical harmonics Got spare money?

quote: Originally posted by KrazyHorse Any good references out there? Basically, my problem is that I have the coefficients of the spherical harmonic decompositon of a function defined on the sphere. The only way I was able to write these down was to define my z axis such that the function was properly azimuthally symmetric. My decomposition therefore has Clm = 0 if m is not 0. What I want to be able to do is transform this decomposition into another basis with new z axis pointing in arbitrary theta, phi direction What I remember of this is that the transformation involves only a sum over m's (not l's) therefore it shouldn't take too much computer time to do (which is good, because I have to transform the bases approximately 3 million times, before doing some other manipulations) this has something to do with the wigner-eckart theorem, but I don't want to think too hard about it. Therefore, the more cookbooky the reference gets the better.

This is another one of your drunk threads right? You're actually asking for the recipe for a martini, but you're too drunk to type, right?

	08-06-2005 03:45									#5
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Du musst erst den Nippel durch die Lasche ziehen...

	08-06-2005 03:47									#6
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Not drunk yet. Might be soon.

Not on martinis, though. Time for more bloody marys

Last night I had a litre and a half of wine but that only kept me buzzing for about 2 hours.

7 or 8 bloody marys ought to do a better job.

	08-06-2005 04:21									#7
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Probably useless, but couldn't you substitute in x-y-z coordinates, do a normal rotation, and then translate back into spherical?

	08-06-2005 04:49									#8
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Uh. Yeah. Extremely useless. That computation takes too long.

In the end IIRC it would be of order (n^3)logn

My way should be n^2

That's the difference between 12 hours on these machines and ~1000 years.

If I was willing to accept a (n^3)logn method then I could have just done it the dumb, straightforward way. (point by point addition)

	08-06-2005 04:54									#9
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The dude who I'm doing work for this summer is going to get his socks blown off tomorrow when I tell him what I'm up to.

	08-06-2005 05:01									#10
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Have you considered a bottom-up approach?

	08-06-2005 05:07									#11
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huh?

	08-06-2005 05:10									#12
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Man, these marys are strong.

Two knocked back and I'm starting to really feel it.

	08-06-2005 05:12									#13
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Anybody who's supposed to be a computer scientist should know this. It seems way to useful for 3-D graphics not to be a standard problem

	08-06-2005 05:14									#14
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In that case maybe I should read it.

	08-06-2005 05:16									#15
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I also had to write my own goddamn sphercial harmonic decomposition program because it doesn't seem to be a standard fpart of an liberary i can find.

bastards.

I used to shudder every time I saw the spherical harmonics in a classas.

	08-06-2005 05:19									#16
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Break this down for me.

You want to transform a matrix?

	08-06-2005 05:19									#17
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What is more scary than a drunk person?




A drunk genius who has diabolical plans to take over the world.

	08-06-2005 05:23									#18
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Are you using (have you considered using) Legendre polynomials?

	08-06-2005 05:30									#19
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I want to transform an orthonormal basis for the set of (complex) scalar functions on the unit sphere

This orthonormal basis is called the spherical harmonics. Each element of this basis is the angular portion of the solution to laplace's equation in spherical coordinates. Think of an expansion in spherical harmonics as being similar to a fourier expansion, except with a different basis set. Fourier expansion is good for describing functions on a lone or a plane. The bessel functions ar good for circles and cyliners. The spherical harmonics are good for spheres.

Since the space is two dimensional there is a 2 dimensional infinite basis set, called the Ylm's (spheriacal hamonics). L despcribles what happens along polar angles. m describes what happens along azimuthatl angles. If there is azimuthal symmetry to your function (rotational symmetry about the z axis), then any coefficient other than the m = 0 ones will be 0, and your Yl0's will look like the legendre polynomials operationg on cos theta (instead of x)

The nice thing about rotations on a function decomposed into spherical harmonic coefficients is that mixing happens only at a single l (i.e. coefficients with different l's have no effect on each other). I just don't know what the transforations look like exactly. I used to , but that was during my graduate QM class (which I took in undergrad about 4 years ago)

	08-06-2005 05:30									#20
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quote: Originally posted by Asher Are you using (have you considered using) Legendre polynomials?

Spherical harmonics are closely related to the (associated) Legendre polynomials

C'm on asher...

	08-06-2005 05:32									#21
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Come on, man, I came up with all this **** by myself.

Somebody has to have done this before me. This **** is an obfvious usage of transformation into an orthionaolrmal basis with nice properties (which have been well-expolored due to hardass pahysicists like Wigner and Dirac)

	08-06-2005 05:33									#22
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No clue bud, never particularly dealt with that.

We don't use spheres in 3D graphics for one thing.

	08-06-2005 05:35									#23
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It seems to me like this woul d be an extremely useful method to rotate objects in 3-D gracphics, sindce any 3-D object (which does not cross itself aong radial lines, and this rpoblem could proabably be solved by decomposing an object into a relatively small number of pieces) can bes describes as a function (distance from arbitrary centre) which depends on theta and phi, and which can thus be decompsed into spherical harmonics.

	08-06-2005 05:40									#24
	Avatar Enlargement: We've got the solution

It doesn't sound particularly efficient.

	08-06-2005 05:41									#25
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Come on, asher.

You don't use spheres in 3-D graphics but the whole point is that your function to be decomposed can despcribe radial distance from an arbitrary centre point.

Like if I wanted to describe an ellipsoid I could write it as r = sqrt(a^2cos^2theta + b^2sin^2theta cos^2 phi + c^2 sin^2 theta sin^2 phi)

other objects can be modelled fairly simply using discrete spherical harmonic demcomposition and then rortated. You have a free hand when it comes to your resolution (depends on how high you go with l), and the object would be automatically smootherd.

	08-06-2005 05:43									#26
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Would have no articficail planes that come from usual vertex decomposition

Everything would be smoothly curved, as you want it to be.

	08-06-2005 05:44									#27
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http://mathworld.wolfram.com/SphericalHarmonic.html

scroll down for images of the first few spherical harmonics.

	08-06-2005 05:44									#28
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Don't you get spherical harmonics in the atoms of a molecule? I know you do get harmonic vibrations between individual atoms of a molecule.

	08-06-2005 05:46									#29
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OF course you do. Wavefucntions of an electron in a coulomb potential give spherical harmonics as solutions.

Harmonic vibrations between atoms is not really relevant to spherical harmonics. That's a radial effect. Spherical harmonics define angeular dependence.

	08-06-2005 05:52									#30
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in other words, s electrons (l = 0) all have angular dependence like Y00 (they are rotationally spymemtric on sphere)

for a p electron, there are 3 choices. They can look like Y1-1, Y10 and Y11 (this is why there are 6 electrons in each with p in each shell; you have to multiply by two bfro the spin degrees of freedom)

for a d electron there are 5 choices Y2-2, Y2-1, Y20, Y21 and Y22

etc.

since the nth electron shell only lallows l values from 0 to n (comes from lapalce's equation) wyou get a total of 2 * (1 + 3 + 5 + ... + (2n+1)) in each shell = 2n^2 electrons, which gives the electronic structiure of the pericodic table.