Spherical Harmonics Rotation Matrices

Rotation of Spherical Harmonics in R^3

 (Franz Zotter)

Rotation Rz around z-axis

Rotation of spherical harmonics around the z-axis, Rz(alpha) is fairly simple, and just follows the trigonometric addition theorems, i.e. the Chebyshev recurrence; which in complex-valued notation corresponds to the powers of exp(i alpha). The evaluation of rotation matrices is most efficient, computing sin(alpha) and cos(alpha) once, and using the addition theorem to build sin(m alpha) and cos(m alpha), in terms of matrix-multiplication. Omitting the dependency of the spherical harmonics on elevation for simplicity, the following equation remains

[ cos(m(phi+alpha)); sin(m(phi+alpha))] = [ cos(alpha), -sin(alpha); sin(alpha), cos(alpha) ]^m [ cos(m phi); sin(m phi)]

Full Rotation R(alpha, beta, gamma)

But Rz(alpha) is not enough to describe all rotations in 3D space. Usually it requires three rotations, e.g.

R(alpha,beta,gamma) = Rz(alpha) Ry(beta) Rz(gamma)

Ry(beta) is not as straight-forward as Rz.

Performing Rotation Ry as Rz and Ry90

But a Ry(beta) can be performed as a rotation Rz(beta) if it is possible to exchange the coordinate axes x and z; or if we know how to rotate by 90 degree Ry90 this looks like

Ry(beta) = Rz90 Ry90 Rz(beta+180) Ry90 Rz90

Full Rotation Performing Ry in terms of Rz and Ry90

R(alpha,beta,gamma) = Rz(alpha+90) Ry90 Rz(beta+180) Ry90 Rz(gamma+90)

Rotation Matrices Ry90 for orders up to the order 21 can be found at http://ambisonics.iem.at/Members/zotter/fixded-angle-90y-rotation-matrices
in the computer-graphics sequence and with condon-shortley phase (see also Zotter's PhD thesis, p36, Figure 9)