Octupoles for octahedral symmetry

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.