# Statistics of ambiguous rotations

## TL;DR

This paper develops statistical methods for analyzing ambiguous rotations, which are rotations known only up to symmetry, with applications in biomechanics, crystallography, and seismology.

## Contribution

It introduces new tests and models for ambiguous rotations, including uniformity tests, parametric models, and regression approaches.

## Key findings

- A test for uniformity of ambiguous rotations is proposed.
- Parametric models for ambiguous rotations are developed.
- An illustrative example with diopside crystal orientations demonstrates the methods.

## Abstract

The orientation of a rigid object can be described by a rotation that transforms it into a standard position. For a symmetrical object the rotation is known only up to multiplication by an element of the symmetry group. Such ambiguous rotations arise in biomechanics, crystallography and seismology. We develop methods for analyzing data of this form. A test of uniformity is given. Parametric models for ambiguous rotations are presented, tests of location are considered, and a regression model is proposed. A brief illustrative example involving orientations of diopside crystals is given.