The ability to describe continuous functions on the space of grain boundary parameters is crucial for investigating the functional relations between the structure and the properties of interfaces, in analogy to the way that continuous distribution functions for orientations (i.e. texture information) have been used extensively in the optimization of polycrystalline microstructures. Here we develop a rigorous framework for the description of continuous functions for a subset of the five-parameter grain boundary space, called the "single-axis grain boundary" space. This space consists of all the boundary plane orientations for misorientations confined to a single axis, and is relevant to the method of presenting boundary plane statistics in widespread current use. We establish the topological equivalence between the single-axis grain boundary space and the 3-sphere, which in turn enables the use of hyperspherical harmonics as basis functions to construct continuous functions. These functions enable the representation of statistical distributions and the construction of functional forms for the structure-property relationships of grain boundaries.
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