Rotation of the spherical harmonics

[KrazyHorse]

reading it now

[Asher]

It's a bigass Spherical Harmonic library coded in C. Hope it helps.

The second link, I mean, not the Sony PDF.

http://www.cs.dartmouth.edu/~geelong/sphere/

SpharmonicKit is a collection of routines, written in C, which implement discrete Legendre and spherical harmonic transforms by a number of different algorithms. For certain algorithms, code for the inverse transform is also provided. Included as well are routines for spherical convolutions.

[KrazyHorse]

okay. didn't have quite what I wanted, but has something equivalently good (can adopt something they said about rotational properties).

[KrazyHorse]

above post refers to the sony PDF

I can use the C library as a reference but I have to do everything in IDL and fortran90 (that's what this research group uses)

[Asher]

http://www.csit.fsu.edu/~burkardt/f_src/polpak/polpak.f90

see the spherical harmonic function

subroutine spherical_harmonic ( l, m, theta, phi, c, s )

!*******************************************************************************
!
!! SPHERICAL_HARMONIC evaluates spherical harmonic functions.
!
!  Discussion:
!
!    The spherical harmonic function Y(L,M,THETA,PHI)(X) is the
!    angular part of the solution to Laplace's equation in spherical
!    coordinates.
!
!    Y(L,M,THETA,PHI)(X) is related to the associated Legendre
!    function as follows:
!
!      Y(L,M,THETA,PHI)(X) = FACTOR * P(L,M)(cos(THETA)) * exp ( i * M * PHI )
!
!    Here, FACTOR is a normalization factor:
!
!      FACTOR = sqrt ( ( 2 * L + 1 ) * ( L - 1 )! / ( 4 * PI * ( L + M )! ) )
!
!    In Mathematica, a spherical harmonic function can be evaluated by
!
!      SphericalHarmonicY [ l, m, theta, phi ]
!
!    Note that notational tradition in physics requires that THETA
!    and PHI represent the reverse of what they would normally mean
!    in mathematical notation; that is, THETA goes up and down, and
!    PHI goes around.
!
!  Modified:
!
!    04 March 2005
!
!  Author:
!
!    John Burkardt
!
!  Reference:
!
!    Milton Abramowitz and Irene Stegun,
!    Handbook of Mathematical Functions,
!    US Department of Commerce, 1964.
!
!    Eric Weisstein,
!    CRC Concise Encyclopedia of Mathematics,
!    CRC Press, 1999.
!
!    Stephen Wolfram,
!    The Mathematica Book,
!    Fourth Edition,
!    Wolfram Media / Cambridge University Press, 1999.
!
!  Parameters:
!
!    Input, integer L, the first index of the spherical harmonic function.
!    Normally, 0 <= L.
!
!    Input, integer M, the second index of the spherical harmonic function.
!    Normally, -L <= M <= L.
!
!    Input, real ( kind = 8 ) THETA, the polar angle, for which
!    0 <= THETA <= PI.
!
!    Input, real ( kind = 8 ) PHI, the longitudinal angle, for which
!    0 <= PHI <= 2*PI.
!
!    Output, real ( kind = 8 ) C(0:L), S(0:L), the real and imaginary
!    parts of the functions Y(L,0:L,THETA,PHI).
!
  implicit none

  integer l

  real ( kind = 8 ) c(0:l)
  integer m
  integer m_abs
  real ( kind = 8 ) phi
  real ( kind = 8 ) plm(0:l)
  real ( kind = 8 ) s(0:l)
  real ( kind = 8 ) theta

  m_abs = abs ( m )

  call legendre_associated_normalized ( l, m_abs, cos ( theta ), plm )

  c(0:l) = plm(0:l) * cos ( real ( m, kind = 8 ) * phi )
  s(0:l) = plm(0:l) * sin ( real ( m, kind = 8 ) * phi )

  if ( m < 0 ) then
    c(0:l) = -c(0:l)
    s(0:l) = -s(0:l)
  end if

  return
end
subroutine spherical_harmonic_values ( n_data, l, m, theta, phi, yr, yi )

!*******************************************************************************
!
!! SPHERICAL_HARMONIC_VALUES returns values of spherical harmonic functions.
!
!  Discussion:
!
!    In Mathematica, the function can be evaluated by
!
!      SphericalHarmonicY [ l, m, theta, phi ]
!
!  Modified:
!
!    05 March 2005
!
!  Author:
!
!    John Burkardt
!
!  Reference:
!
!    Milton Abramowitz and Irene Stegun,
!    Handbook of Mathematical Functions,
!    US Department of Commerce, 1964.
!
!    Eric Weisstein,
!    CRC Concise Encyclopedia of Mathematics,
!    CRC Press, 1998.
!
!    Stephen Wolfram,
!    The Mathematica Book,
!    Fourth Edition,
!    Wolfram Media / Cambridge University Press, 1999.
!
!  Parameters:
!
!    Input/output, integer N_DATA.
!    On input, if N_DATA is 0, the first test data is returned, and
!    N_DATA is set to the index of the test data.  On each subsequent
!    call, N_DATA is incremented and that test data is returned.  When
!    there is no more test data, N_DATA is set to 0.
!
!    Output, integer L, integer M, real ( kind = 8 ) THETA, PHI, the arguments
!    of the function.
!
!    Output, real ( kind = 8 ) YR, YI, the real and imaginary parts of
!    the function.
!
  implicit none

  integer, parameter :: nmax = 20

  integer l
  integer, save, dimension ( nmax ) :: l_vec = (/ &
     0,  1,  2,  &
     3,  4,  5,  &
     5,  5,  5,  &
     5,  4,  4,  &
     4,  4,  4,  &
     3,  3,  3,  &
     3,  3 /)
  integer m
  integer, save, dimension ( nmax ) :: m_vec = (/ &
     0,  0,  1,  &
     2,  3,  5,  &
     4,  3,  2,  &
     1,  2,  2,  &
     2,  2,  2,  &
    -1, -1, -1,  &
    -1, -1 /)
  integer n_data
  real ( kind = 8 ) phi
  real ( kind = 8 ), save, dimension ( nmax ) :: phi_vec = (/ &
    0.1047197551196598D+01, 0.1047197551196598D+01, 0.1047197551196598D+01, &
    0.1047197551196598D+01, 0.1047197551196598D+01, 0.6283185307179586D+00, &
    0.6283185307179586D+00, 0.6283185307179586D+00, 0.6283185307179586D+00, &
    0.6283185307179586D+00, 0.7853981633974483D+00, 0.7853981633974483D+00, &
    0.7853981633974483D+00, 0.7853981633974483D+00, 0.7853981633974483D+00, &
    0.4487989505128276D+00, 0.8975979010256552D+00, 0.1346396851538483D+01, &
    0.1795195802051310D+01, 0.2243994752564138D+01 /)
  real ( kind = 8 ) theta
  real ( kind = 8 ), save, dimension ( nmax ) :: theta_vec = (/ &
    0.5235987755982989D+00, 0.5235987755982989D+00, 0.5235987755982989D+00, &
    0.5235987755982989D+00, 0.5235987755982989D+00, 0.2617993877991494D+00, &
    0.2617993877991494D+00, 0.2617993877991494D+00, 0.2617993877991494D+00, &
    0.2617993877991494D+00, 0.6283185307179586D+00, 0.1884955592153876D+01, &
    0.3141592653589793D+01, 0.4398229715025711D+01, 0.5654866776461628D+01, &
    0.3926990816987242D+00, 0.3926990816987242D+00, 0.3926990816987242D+00, &
    0.3926990816987242D+00, 0.3926990816987242D+00 /)
  real ( kind = 8 ) yi
  real ( kind = 8 ), save, dimension ( nmax ) :: yi_vec = (/ &
    0.0000000000000000D+00,  0.0000000000000000D+00, -0.2897056515173922D+00, &
    0.1916222768312404D+00,  0.0000000000000000D+00,  0.0000000000000000D+00, &
    0.3739289485283311D-02, -0.4219517552320796D-01,  0.1876264225575173D+00, &
   -0.3029973424491321D+00,  0.4139385503112256D+00, -0.1003229830187463D+00, &
    0.0000000000000000D+00, -0.1003229830187463D+00,  0.4139385503112256D+00, &
   -0.1753512375142586D+00, -0.3159720118970196D+00, -0.3940106541811563D+00, &
   -0.3940106541811563D+00, -0.3159720118970196D+00 /)
  real ( kind = 8 ) yr
  real ( kind = 8 ), save, dimension ( nmax ) :: yr_vec = (/ &
   0.2820947917738781D+00,  0.4231421876608172D+00, -0.1672616358893223D+00, &
  -0.1106331731112457D+00,  0.1354974113737760D+00,  0.5390423109043568D-03, &
  -0.5146690442951909D-02,  0.1371004361349490D-01,  0.6096352022265540D-01, &
  -0.4170400640977983D+00,  0.0000000000000000D+00,  0.0000000000000000D+00, &
   0.0000000000000000D+00,  0.0000000000000000D+00,  0.0000000000000000D+00, &
   0.3641205966137958D+00,  0.2519792711195075D+00,  0.8993036065704300D-01, &
  -0.8993036065704300D-01, -0.2519792711195075D+00 /)

  if ( n_data < 0 ) then
    n_data = 0
  end if

  n_data = n_data + 1

  if ( nmax < n_data ) then
    n_data = 0
    l = 0
    m = 0
    theta = 0.0D+00
    phi = 0.0D+00
    yr = 0.0D+00
    yi = 0.0D+00
  else
    l = l_vec(n_data)
    m = m_vec(n_data)
    theta = theta_vec(n_data)
    phi = phi_vec(n_data)
    yr = yr_vec(n_data)
    yi = yi_vec(n_data)
  end if

  return
end

[The Vagabond]

Originally posted by 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.

Good for circles and cylinders are just the ordinary sines and cosines (Fourier series), smartass. The Bessel functions take care of what happens in the radial direction from a circle/cylinder. But they have nothing to do with what happens on circles or cylinders.

[Ramo]

Unless you've got a funky metric.

[The Vagabond]

Yeah, after a couple of bottles of wine your metric may become funky indeed.

[KrazyHorse]

I didn't remember the proper solutions to the Laplace equation in cylindrical coordinates while drunk. So shoot me...