AI 中文总结
该研究针对实特殊正交群SO₄(ℝ)上的有理曲线，受运动多项式分解等启发，通过Cayley分解等方法构造性地完成Kempe分解，所得算法显式并附实例说明。
AI 中文摘要
我们研究实特殊正交群SO₄(ℝ)上有理曲线的构造性Kempe分解。受运动多项式分解及实经典群上有理矩阵曲线的启发，我们证明：对于每个次数为2d（d≥1）的有理曲线，其可先分解为d个二次有理曲线，再分解为至多2d个平面旋转曲线的乘积，其中每个平面旋转曲线逐点固定一个二维平面。该构造通过Cayley分解提取左右迷向多项式部分，分解对应的四元数多项式，并将线性因子与等范数多项式配对实现。所得算法是显式的，并用实例说明。
英文摘要
We study constructive Kempe factorizations for rational curves on $\operatorname{SO}_4(\mathbb{R})$. Motivated by motion-polynomial factorization and rational matrix curves on real classical groups, we prove that every rational curve of degree $2d$ with $d\ge1$ 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.
