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

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• 作者：Rida T. Farouki ; Graziano Gentili…
• 关键词：Pythagorean–hodograph curves ; Rotation–minimizing frames ; Quaternion polynomials ; Rotation indicatrix
• 刊名：Advances in Computational Mathematics
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：43
• 期：1
• 页码：1-24
• 全文大小：
• 刊物类别：Computer Science
• 刊物主题：Computational Mathematics and Numerical Analysis; Mathematical Modeling and Industrial Mathematics; Mathematical and Computational Biology; Computational Science and Engineering; Visualization;
• 出版者：Springer US
• ISSN：1572-9044
• 卷排序：43

文摘

A rotation–minimizing frame (f₁,f₂,f₃) on a space curve r(ξ) defines an orthonormal basis for \(\mathbb {R}^{3}\) in which \(\mathbf {f}_{1}=\mathbf {r}^{\prime }/|\mathbf {r}^{\prime }|\) is the curve tangent, and the normal–plane vectors f₂, f₃ exhibit no instantaneous rotation about f₁. 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 \(\mathbf {r}^{\prime }(\xi )=\mathcal {A}(\xi )\,\mathbf{i} \,\mathcal {A}^{*}(\xi )\) for some quaternion polynomial \(\mathcal {A}(\xi )\). By introducing the notion of the rotation indicatrix and the core of the quaternion polynomial \(\mathcal {A}(\xi )\), 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.