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.

翻译：四次球谐函数因其能够再现八面体对称性而被广泛应用于体积框架场设计中。本文展示如何利用三次谐波（八极子）实现相同目的，从而减少参数数量和计算复杂度。该方法的关键要素包括：● 八极子流形的隐式方程（该类八极子具有至多相差-1倍乘的八面体对称性），● 相应的旋转不变度量（用于衡量八极子与特定对称性的偏离程度），以及● 在场优化过程中补偿八极子对称性缺失的光滑惩罚项。