# Minkowski products of unit quaternion sets

## TL;DR

This paper introduces and analyzes the Minkowski product of unit quaternion sets to characterize combined spatial rotations with uncertainties, providing new mathematical insights and visualization methods.

## Contribution

It presents the first analysis of Minkowski products of unit quaternion sets, including closure conditions, specific set products, and visualization techniques.

## Key findings

- Closure under Minkowski product for spherical caps on $S^3$
- Methods for visualizing quaternion sets in $
r^3$
- Principles for identifying boundary points in Minkowski products

## Abstract

The Minkowski product of unit quaternion sets is introduced and analyzed, motivated by the desire to characterize the overall variation of compounded spatial rotations that result from individual rotations subject to known uncertainties in their rotation axes and angles. For a special type of unit quaternion set, the spherical caps of the 3-sphere $S^3$ in $\mathbb{R}^4$, closure under the Minkowski product is achieved. Products of sets characterized by fixing either the rotation axis or rotation angle, and allowing the other to vary over a given domain, are also analyzed. Two methods for visualizing unit quaternion sets and their Minkowski products in $\mathbb{R}^3$ are also discussed, based on stereographic projection and the Lie algebra formulation. Finally, some general principles for identifying Minkowski product boundary points are discussed in the case of full-dimension set operands.