多项式曲线的有理旋转最小框架的全面特征化
A comprehensive characterization of the set of polynomial curves with rational rotation-minimizing frames
AI总结：
本文研究了具有有理旋转最小框架（RRMF）的多项式曲线，通过引入旋转指示量和四元数多项式核心概念，全面特征化了RRMF曲线空间，区分了空间RRMF曲线与平面情况，为新算法提供基础。
摘要：
在空间曲线r(ξ)上定义的旋转最小框架（f1,f2,f3）为R³的正交基，在此基中，f1=r′/|r′|为曲线切线，而法平面向量f2,f3关于f1无瞬时旋转。允许有理旋转最小框架（RRMF）的多项式曲线是Pythagorean-hodograph（PH）曲线的子集，由积分形式r′(ξ)=A(ξ)iA*(ξ)（其中A(ξ)为四元数多项式）指定。通过引入旋转指示量和四元数多项式A(ξ)的核心概念，发展了RRMF曲线的完整空间特征化，涵盖所有已知的特殊情形。这一新特征化有助于澄清RRMF曲线完整空间的结构，区分空间RRMF曲线与平凡（平面）情况，并为新构造算法铺平道路。
英文摘要：
A rotation-minimizing frame $({\bf f}_1,{\bf f}_2,{\bf f}_3)$ on a space curve ${\bf r}(ξ)$ defines an orthonormal basis for $\mathbb{R}^3$ in which ${\bf f}_1={\bf r}'/|{\bf r}'|$ is the curve tangent, and the normal-plane vectors ${\bf f}_2$, ${\bf f}_3$ exhibit no instantaneous rotation about ${\bf f}_1$. Polynomial curves that admit rational rotation-minimizing frames (or RRMF curves) form a subset of the Pythagorean-hodograph (PH) curves, specified by integrating the form ${\bf r}'(ξ)={\cal A}(ξ)\,{\bf i}\,{\cal A}^*(ξ)$ for some quaternion polynomial ${\cal A}(ξ)$. By introducing the notion of rotation indicatrix and of core of the quaternion polynomial ${\cal A}(ξ)$, a comprehensive characterization of the complete space of RRMF curves is developed, that subsumes all previously known special cases. This novel characterization helps clarify the structure of the complete space of RRMF curves, distinguishes the spatial RRMF curves from trivial (planar) cases, and paves the way toward new construction algorithms.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
