An Algorithm for the Factorization of Split Quaternion Polynomials

TL;DR

This paper introduces an algorithm for factorizing split quaternion polynomials into linear factors, with geometric interpretations and applications to rational motions in hyperbolic and Euclidean kinematics.

Contribution

The paper presents a novel algorithm for factorizing split quaternion polynomials and extends techniques to dual quaternions for motion analysis.

Findings

Algorithm successfully factorizes split quaternion polynomials when possible.

Geometric interpretation relates to rulings on a quadric of non-invertible quaternions.

Method extends to factorization of motion polynomials in kinematics.

Abstract

We present an algorithm to compute all factorizations into linear factors of univariate polynomials over the split quaternions, provided such a factorization exists. Failure of the algorithm is equivalent to non-factorizability for which we present also geometric interpretations in terms of rulings on the quadric of non-invertible split quaternions. However, suitable real polynomial multiples of split quaternion polynomials can still be factorized and we describe how to find these real polynomials. Split quaternion polynomials describe rational motions in the hyperbolic plane. Factorization with linear factors corresponds to the decomposition of the rational motion into hyperbolic rotations. Since multiplication with a real polynomial does not change the motion, this decomposition is always possible. Some of our ideas can be transferred to the factorization theory of motion polynomials. These are polynomials over the dual quaternions with real norm polynomial and they describe rational motions in Euclidean kinematics. We transfer techniques developed for split quaternions to compute new factorizations of certain dual quaternion polynomials.