Rotate orthogonal spherical-cap Slepian functions centered at the North pole to a different location.
Usage
tapersrot = SHRotateTapers(tapers, taper_order, nrot, x, dj)
Returns
- tapersrot : float, dimension ((lmax+1)**2, nrot)
- An array containing the spherical harmonic coefficients of the rotated spherical-cap functions, where lmax is the spherical harmonic bandwidth of the functions. Each column corresponds to a single function of which the spherical harmonic coefficients can be unpacked with SHVectorToCilm.
Parameters
- tapers : float, dimension (lmax+1, nrot)
- An array containing the eigenfunctions of the spherical-cap concentration problem obtained from SHReturnTapers. The functions are listed by columns, ordered from best to worst concentrated.
- taper_order : integer, dimension (nrot)
- The angular order of the non-zero spherical harmonic coefficients in each column of tapers.
- nrot : integer
- The number of functions to rotate, which must be less than or equal to (lmax+1)**2.
- x : float, dimension(3)
- The three Euler angles, alpha, beta, and gamma, in radians.
- dj : float, dimension (lmax+1, lmax+1, lmax+1)
- The rotation matrix dj(pi/2), obtained from a call to djpi2.
Description
SHRotateTapers will rotate a set of orthogonal spherical-cap Slepian functions originally centered at the North pole to a different location according to the three Euler angles in the vector x. The original matrix tapers is computed by a call to SHReturnTapers. Only the first nrot tapers are rotated, each of which is returned in a column of the output matrix tapersrot. The spherical harmonic coefficients are geodesy 4pi normalized, and each column of tapersrot can be unpacked using SHVectorCilm. 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 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.
This routine first converts the real coefficients to complex form using SHrtoc. Then the coefficients are converted to indexed form using SHCilmToCindex, these are sent to SHRotateCoef, the result if converted back to cilm complex form using SHCindexToCilm, and these are finally converted back to real form using SHctor.
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