A comprehensive characterization of the set of polynomial curves with rational rotation-minimizing frames

TL;DR

This paper provides a complete mathematical characterization of polynomial curves with rational rotation-minimizing frames, expanding understanding and enabling new construction methods for these special space curves.

Contribution

It introduces the notions of rotation indicatrix and core of quaternion polynomials to fully characterize RRMF curves, unifying previous special cases.

Findings

Develops a comprehensive mathematical framework for RRMF curves.

Distinguishes spatial RRMF curves from planar cases.

Facilitates new algorithms for constructing RRMF curves.

Abstract

A rotation-minimizing frame $({\bf f}_1,{\bf f}_2,{\bf f}_3)$ on a space curve ${\bf r}(\xi)$ 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 ${\bf r}'(\xi)={\cal A}(\xi)\,{\bf i}\,{\cal A}^*(\xi)$ for some quaternion polynomial ${\cal A}(\xi)$. By introducing the notion of rotation indicatrix and of core of the quaternion polynomial ${\cal A}(\xi)$, 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.