A Harmonic Framework for Vector Fields and Differential Operators on SO(3)

Abstract

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.