# 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)