Re: orthogonal functions from normalized standing wave functions
- To: mathgroup at smc.vnet.net
- Subject: [mg68038] Re: orthogonal functions from normalized standing wave functions
- From: Roger Bagula <rlbagula at sbcglobal.net>
- Date: Fri, 21 Jul 2006 05:37:29 -0400 (EDT)
- References: <e92eja$lte$1@smc.vnet.net> <e9frat$2jv$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
The relationship of spherical harmonics to areas like projective planes
isn't obvious.
I noticed this several years ago and thought others should be aware of
this relationship!
Mathematica:
FullSimplify[ComplexExpand[{-Re[SphericalHarmonicY[2,
1, p, t]], -Im[SphericalHarmonicY[2, 1, p, t]], Im[
SphericalHarmonicY[2, 2, p, t]]}]/( Sqrt[15/(2*Pi)]/4)]
ParametricPlot3D[{Re[SphericalHarmonicY[2,
1, p, t]], Im[SphericalHarmonicY[2, 1, p, t]], Im[
SphericalHarmonicY[2, 2,
p, t]]}/( Sqrt[15/(2*Pi)]/4), {p, 0, Pi}, {t, 0, Pi}]
(* so(3) like matrix wit {x,y,z} a sphere*)
M = {{0, Sin[p]*Sin[t], -Sin[p]*Cos[t]},
{-Sin[p]*Sin[t], 0, Cos[p]},
{Sin[p]*Cos[t], -Cos[p], 0}}
M2 = FullSimplify[2*M.M]
MatrixForm[M2]
(* picking out the spherical harmonics*)
{M2[[1, 2]], M2[[1, 3]], M2[[2, 3]]}
ParametricPlot3D[{M2[[1, 2]], M2[[1, 3]], M2[[2, 3]]}, {p, 0, Pi}, {t,
0, Pi}]