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Kempe factorizations for rational curves on SO4(ℝ)
Abstract
We study constructive Kempe factorizations for rational curves on SO4(ℝ). Motivated by motion-polynomial factorization and rational matrix curves on real classical groups, we prove that every rational curve of degree 2d with d≥1 first factors into d quadratic rational curves and then into a product of at most 2d planar rotation curves, where each planar rotation curve fixes a two-dimensional plane pointwise. The construction proceeds by extracting left and right isoclinic polynomial parts via Cayley's factorization, factoring the corresponding quaternion polynomials, and pairing linear factors with equal norm polynomials. The resulting algorithms are explicit and are illustrated by examples.
Related
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BibTeX
@misc{li2026kempe,
title = {{Kempe factorizations for rational curves on $\operatornameSO_4(\mathbbR)$}},
author = {Yifan Li and Zijia Li and Ke Ye},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.30577},
eprint = {2608.30577},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30577},
}
RIS
TY - UNPB
TI - Kempe factorizations for rational curves on SO_4(ℝ)
AU - Yifan Li
AU - Zijia Li
AU - Ke Ye
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.30577
UR - https://arxiv.org/abs/2608.30577
AB - We study constructive Kempe factorizations for rational curves on SO_4(ℝ). Motivated by motion-polynomial factorization and rational matrix curves on real classical groups, we prove that every rational curve of degree 2d with d≥1 first factors into d quadratic rational curves and then into a product of at most 2d planar rotation curves, where each planar rotation curve fixes a two-dimensional plane pointwise. The construction proceeds by extracting left and right isoclinic polynomial parts via Cayley's factorization, factoring the corresponding quaternion polynomials, and pairing linear factors with equal norm polynomials. The resulting algorithms are explicit and are illustrated by examples.
ER -