Quadratic Motion Polynomials With Irregular Factorizations
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| author | Thimm, Daren A. Li, Zijia Schröcker, Hans-Peter Siegele, Johannes |
| author_facet | Thimm, Daren A. Li, Zijia Schröcker, Hans-Peter Siegele, Johannes |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_08350 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quadratic Motion Polynomials With Irregular Factorizations Thimm, Daren A. Li, Zijia Schröcker, Hans-Peter Siegele, Johannes Rings and Algebras 12D05 15A66 70B10 16S36 30C15 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. |
| title | Quadratic Motion Polynomials With Irregular Factorizations |
| topic | Rings and Algebras 12D05 15A66 70B10 16S36 30C15 |
| url | https://arxiv.org/abs/2504.08350 |