Quadratic Motion Polynomials with Irregular Factorizations

Thimm, Daren A.

doi:10.1007/s00006-025-01426-2

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Quadratic Motion Polynomials with Irregular Factorizations

2026-01-24

Thimm, Daren A.

Li, Zijia

Schrocker, Hans-Peter *

Siegele, Johannes

DOI：10.1007/s00006-025-01426-2

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Abstract

En 中文

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.

Keywords:

Conformal geometric algebra

Conformal kinematics

Rational motion

Motion factorization

Villarceau motion

Circular translation