Symmetries of Spherical Harmonics: applications to ambisonics
Michael Chapman
Spherical harmonics are often represented as images of three dimensional forms. Inspection shows that mirroring those (here, a reflection in the planes of the three axis pairs) can result in no change. That is symmetries exist.
Using direction cosine formulae for the spherical harmonics these symmetries are established and then extrapolated to higher degrees.
The results are used
- to produce formulae generalised for an any order signal set for reflection through the planes
- to produce 'skeleton' matrices for pitch and roll rotations in 90° steps, for any ambisonic order, and
- to algebraicly further explore 'dominance'.
