# Quadratic Motion Polynomials with Irregular Factorizations

## TL;DR

This paper explores special types of motion polynomials that can be factored in non-standard ways, revealing new algebraic properties and examples.

## Contribution

The paper characterizes quadratic motion polynomials with irregular factorizations using algebraic equations and provides new examples and sub-case analyses.

## Key findings

- Quadratic motion polynomials with irregular factorizations are characterized via algebraic equations.
- Examples show these polynomials can have one to infinitely many unique factorizations.
- Special sub-cases like conformal Villarceau motion and circular translation are uniquely identified.

## Abstract

Motion polynomials are a specific type of polynomial over a Clifford algebra that can conveniently describe rational motions. There exists an algorithm for the factorization of motion polynomials that works in generic cases. It hinges on the invertibility of a certain coefficient occurring in the algorithm. If this coefficient is not invertible, factorizations may or may not exist. In the case of existence we call this an irregular factorization. We characterize quadratic motion polynomials with irregular factorizations in terms of algebraic equations and present examples whose number of unique factorizations range from one to infinitely many. For two special sub-cases we show the unique existence of such polynomials. In case of commuting factors we obtain the conformal Villarceau motion, in case of rigid body motions the circular translation.