Determine the spherical harmonic coefficients of a real function expressed in complex harmonics rotated by three Euler angles.

Usage

rcoef = SHRotateCoef (x, coef, dj, [lmax])

Returns

rcoef : flaot, dimension (2, (lmax+1)*(lmax+2)/2)
The spherical harmonic coefficients of the rotated function in indexed form.

Parameters

x : float, dimension(3)
The three Euler angles, alpha, beta, and gamma, in radians.
coef : float, dimension (2, (lmaxin+1)*(lmaxin+2)/2)
The input complex spherical harmonic coefficients of the real function. This is an indexed array where the real and complex components are given by coef[0,:] and coef[1,:], respectively. The functions SHCilmToCindex and SHCindexToCilm are used to convert to and from indexed and cilm[:,:,:] form. The coefficients must correspond to unit-normalized spherical harmonics that possess the Condon-Shortley phase convention.
dj : float, dimension (lmaxin2+1, lmaxin2+1, lmaxin2+1)
The rotation matrix dj(pi/2) obtained from a call to djpi2.
lmax : optional, integer, default = lmaxin
The maximum spherical harmonic degree of the input and output coefficients.

Description

SHRotateCoef will take the complex spherical harmonic coefficients of a real function, rotate it according to the three Euler anlges in x, and output the spherical harmonic coefficients of the rotated function. The input and output coefficients are in an indexed form that can be converted to and from cilm[:,:,:] form by using the functions SHCilmToCindex and SHCindexToCilm. The coefficients must correspond to unit-normalized spherical harmonics that possess the Condon-Shortley phase convention. Real spherical harmonics can be converted to and from complex form using SHrtoc and SHctor. The input rotation matrix dj is computed by a call to djpi2.

The rotation of a coordinate system or body can be viewed in two complementary ways involving three successive rotations. Both methods have the same initial and final configurations, and the angles listed in both schemes are the same. This routine uses the ‘y convention’, where the second rotation axis corresponds to the y axis.

Scheme A:

(I) Rotation about the z axis by alpha. (II) Rotation about the new y axis by beta. (III) Rotation about the new z axis by gamma.

Scheme B:

(I) Rotation about the z axis by gamma. (II) Rotation about the initial y axis by beta. (III) Rotation about the initial z axis by alpha.

The rotations can further be viewed either as a rotation of the coordinate system or the physical body. For a rotation of the coordinate system without rotation of the physical body, use

x(alpha, beta, gamma).

For a rotation of the physical body without rotation of the coordinate system, use

x(-gamma, -beta, -alpha).

The inverse transform of x(alpha, beta, gamma) is x(-gamma, -beta, -alpha).

Note that this routine uses the “y convention”, where the second rotation is with respect to the new y axis. If alpha, beta, and gamma were originally defined in terms of the “x convention”, where the second rotation was with respect to the new x axis, the Euler angles according to the y convention would be alpha_y=alpha_x-pi/2, beta_x=beta_y, and gamma_y=gamma_x+pi/2.

See also

djpi2, shrotaterealcoef, shctor, shrtoc, shcilmtocindex, shcindextocilm

Tags: python
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