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

作者：Farouki Rida T^*; Gentili Graziano; Giannelli Carlotta; Sestini Alessandra; Stoppato Caterina

来源：Advances in Computational Mathematics, 2017, 43(1): 1-24.

DOI：10.1007/s10444-016-9473-0

• Pythagorean-hodograph curves
• Rotation-minimizing frames
• Quaternion polynomials
• Rotation indicatrix

A rotation-minimizing frame (f (1),f (2),f (3)) on a space curve r(xi) defines an orthonormal basis for in which is the curve tangent, and the normal-plane vectors f (2), f (3) exhibit no instantaneous rotation about 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 for some quaternion polynomial . By introducing the notion of the rotation indicatrix and the core of the quaternion polynomial , 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.

• 出版日期2017-2