On spherical harmonics possessing octahedral symmetry

TL;DR

This paper derives implicit equations for a special class of real spherical harmonics with octahedral symmetry and introduces a measure of deviation from this symmetry, aiding in directional fields design.

Contribution

It provides explicit equations and a rotationally invariant deviation measure for spherical harmonics with octahedral symmetry, a novel approach for symmetry analysis.

Findings

Derived implicit equations for octahedral symmetric spherical harmonics

Constructed a rotationally invariant deviation measure

Applied to directional fields design in 3D space

Abstract

In this paper, we present the implicit equations for one special class of real-valued spherical harmonics with octahedral symmetry. Based on this representation, we construct the rotationally invariant measure of deviation from the specified symmetry. The spherical harmonics we consider have some applications in the area of directional fields design due to their ability to represent mutually orthogonal axes in 3D space, not relative to their order and orientation.