Spherical harmonics provide a smooth, orthogonal, and symmetry-adapted basis to expand functions on a sphere, and they are used routinely in computer graphics, signal processing and different fields of science, from geology to quantum chemistry. More recently, spherical harmonics have become a key component of rotationally equivariant models for geometric deep learning, where they are used in combination with distance-dependent functions to describe the distribution of neighbors within local spherical environments within a point cloud. We present a fast and elegant algorithm for the evaluation of the real-valued spherical harmonics. Our construction integrates many of the desirable features of existing schemes and allows to compute Cartesian derivatives in a numerically stable and computationally efficient manner. We provide an efficient C implementation of the proposed algorithm, along with easy-to-use Python bindings.

翻译：球谐函数为球面上的函数展开提供了平滑、正交且对称适应的基函数，被广泛应用于计算机图形学、信号处理以及从地质学到量子化学等不同科学领域。近期，球谐函数已成为几何深度学习中等变旋转模型的关键组成部分，在这些模型中，它们与距离依赖函数结合使用，以描述点云内局部球面环境中邻居的分布。我们提出了一种快速且优雅的实值球谐函数计算算法。我们的构造方法融合了现有方案的诸多优良特性，能够以数值稳定且计算高效的方式求解笛卡尔坐标系下的导数。我们提供了所提算法的高效C语言实现，并附有易于使用的Python接口。