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SHRotateCoef

SHRotateCoef -

Determine the spherical harmonic coefficients of a complex function rotated by three Euler angles.

SYNOPSIS

SUBROUTINE SHRotateCoef (

X, COEF, RCOEF, DJ, LMAX )

    REAL*8

    X(3), COEF(2, (LMAX+1)*(LMAX+2)/2), RCOEF(2, (LMAX+1)*(LMAX+2)/2), DJ(LMAX+1, LMAX+1, LMAX+1)

    INTEGER

    LMAX

DESCRIPTION

SHRotateCoef will take the complex spherical harmonic coefficients of a 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(2,:,:) 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.

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).

To perform the inverse transform of X(alpha, beta, gamma), use 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 orginally defined in terms of the "x convention", where the second rotation was with respect to the newx 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.

ARGUMENTS

X

(input) REAL*8, DIMENSION(3 )

The three Euler angles, alpha, beta, and gamma.

COEF

(input) REAL*8, DIMENSION (2, (LMAX+1)*(LMAX+2)/2)

The input complex spherical harmonic coefficients. This is an indexed array where the real and complex components are given by COEF(1,:) and COEF(2,:), respectively. The functions SHCilmToCindex and SHCindexToCilm are used to convert to and from indexed and CILM(2,:,:) form. The coefficients must correspond to unit-normalized spherical harmonics that possess the Condon-Shortley phase convention.

RCOEF

(output) REAL*8, DIMENSION (2, (LMAX+1)*(LMAX+2)/2)

The spherical harmonic coefficients of the rotated function in indexed form.

DJ

(input) REAL*8, DIMENSION (LMAX+1, LMAX+1, LMAX+1)

The rotation matrix DJ(pi/2), obtained from a call to DJPi2.

LMAX

(input) INTEGER, OPTIONAL

The maximum spherical harmonic degree of the input and output coefficients.

NOTES

This routine is based upon code originally written by Guy Masters, which was then subsequently modified by Mark Simons and myself.

SEE ALSO

djpi2(1), shrotaterealcoef(1), shctor(1), shrtoc(1), shcilmtocindex(1), shcindextocilm(1)

http://shtools.ipgp.fr/

COPYRIGHT AND LICENSE

Copyright 2012 by Mark Wieczorek <wieczor@ipgp.fr>.

This is free software; you can distribute and modify it under the terms of the revised BSD license.