Rotation of the spherical harmonics

[Asher]

Break this down for me.

You want to transform a matrix?

[MrFun]

What is more scary than a drunk person?




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

[Asher]

Are you using (have you considered using) Legendre polynomials?

[KrazyHorse]

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)

[KrazyHorse]

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...

[KrazyHorse]

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)

[Asher]

No clue bud, never particularly dealt with that.

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

[KrazyHorse]

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.

[Asher]

It doesn't sound particularly efficient.

[KrazyHorse]

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.

[KrazyHorse]

Would have no articficail planes that come from usual vertex decomposition

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

[KrazyHorse]

http://mathworld.wolfram.com/SphericalHarmonic.html

scroll down for images of the first few spherical harmonics.

[Dinner]

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.

[KrazyHorse]

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.

[KrazyHorse]

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.