Octupoles for octahedral symmetry

TL;DR

This paper introduces a novel method using degree 3 spherical harmonics (octupoles) to achieve octahedral symmetry in volumetric frame fields, reducing complexity compared to traditional degree 4 harmonics.

Contribution

It demonstrates how to utilize octupoles for octahedral symmetry, providing implicit equations, invariant measures, and smoothing techniques to improve field design.

Findings

Reduces parameters and computational complexity in symmetry enforcement.

Provides explicit equations and measures for octupoles with octahedral symmetry.

Enhances volumetric frame field design with a novel harmonic approach.

Abstract

Spherical harmonics of degree 4 are widely used in volumetric frame fields design due to their ability to reproduce octahedral symmetry. In this paper we show how to use harmonics of degree 3 (octupoles) for the same purpose, thereby reducing number of parameters and computational complexity. The key ingredients of the presented approach are \quad \textbullet \ implicit equations for the manifold of octupoles possessing octahedral symmetry up to multiplication by $-1$, \quad \textbullet \ corresponding rotationally invariant measure of octupole's deviation from the specified symmetry, \quad \textbullet \ smoothing penalty term compensating the lack of octupoles' symmetries during a field optimization.