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Quadratic Motion Polynomials with Irregular Factorizations.

Daren A Thimm1, Zijia Li2,3, Hans-Peter Schröcker1

  • 1Department of Basic Sciences in Engineering Sciences, University of Innsbruck, Technikerstraße 16, 6020 Innsbruck, Austria.

Advances in Applied Clifford Algebras
|January 27, 2026
PubMed
Summary
This summary is machine-generated.

This study characterizes irregular factorizations of quadratic motion polynomials, which describe rational motions using Clifford algebra. It identifies conditions for unique and multiple factorizations, revealing connections to conformal Villarceau and circular translations.

Keywords:
Circular translationConformal geometric algebraConformal kinematicsMotion factorizationRational motionVillarceau motion

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Area of Science:

  • Algebraic Geometry
  • Geometric Algebra
  • Robotics

Background:

  • Motion polynomials, a class of polynomials over Clifford algebras, are used to represent rational motions.
  • A standard algorithm exists for factoring motion polynomials, but it fails when a key coefficient is non-invertible, leading to irregular factorizations.

Purpose of the Study:

  • To characterize quadratic motion polynomials that admit irregular factorizations.
  • To analyze the conditions leading to the existence and number of unique factorizations in these irregular cases.

Main Methods:

  • Characterization of irregular factorizations using algebraic equations.
  • Analysis of coefficient invertibility in the motion polynomial factorization algorithm.
  • Examination of specific sub-cases, including commuting factors and rigid body motions.

Main Results:

  • Quadratic motion polynomials with irregular factorizations are fully characterized by algebraic conditions.
  • Examples demonstrate scenarios with one to infinitely many unique factorizations.
  • Two special sub-cases yield unique polynomial factorizations: conformal Villarceau motion (commuting factors) and circular translation (rigid body motions).

Conclusions:

  • The study provides a comprehensive understanding of irregular factorizations in motion polynomials.
  • These findings offer insights into the geometric interpretations of specific motions derived from algebraic structures.
  • The characterization aids in analyzing and constructing rational motions with specific properties.