AI 中文总结
该研究针对SO(3)提出调和框架，采用左右不变标架避免数值不稳定性，推导微分算子频域公式并保持带限性，为MTEX提供数学基础。
AI 中文摘要
我们利用调和级数展开，给出了旋转群SO(3)上切向量场与微分算子的完整框架，为其在晶体学织构分析软件MTEX中的实现提供了数学基础。核心思路是采用标准左不变和右不变标架作为切丛的全局标准正交标架，从而避免了由欧拉角参数化的雅可比矩阵导出的经典切空间基的数值不稳定性。虽然标架的选择主要由其几何与数值特性驱动，但其全部潜力在调和框架中得以体现：通过切向量场的标架分量的调和展开来表示切向量场，我们推导了梯度、散度和旋度的显式频域公式，使这些算子可直接应用于调和系数；此外，我们证明这些微分算子保持调和带限性。另外，切向量场的左不变与右不变表示可在频域中直接相互转换，且调和带宽的增加最多为1阶。
英文摘要
We present a comprehensive framework for tangent vector fields and differential operators on the rotation group $\mathrm{SO}(3)$ using harmonic series expansions, which provides a mathematical foundation for their implementation in the crystallographic texture analysis software MTEX. The central idea is to employ the standard left- and right-invariant frames as global orthonormal frames of the tangent bundle, thereby avoiding the numerical instabilities of the classical tangent space basis derived from the Jacobian of the Euler angle parametrization. While the choice of frames is primarily motivated by their geometric and numerical properties, their full potential emerges in the harmonic setting. Representing tangent vector fields through harmonic expansions of their frame components, we derive explicit frequency domain formulas for the gradient, divergence, and curl, allowing these operators to be applied directly to the harmonic coefficients. Moreover, we show that these differential operators preserve harmonic band-limitedness. In addition, the left- and right-invariant representations of tangent vector fields can be transformed into one another directly in the frequency domain, with an increase in harmonic bandwidth of at most one degree.
Comments17 pages, 3 figures
