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Advances in Computational Mathematics

, Volume 43, Issue 1, pp 1–24 | Cite as

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

  • Rida T. Farouki
  • Graziano Gentili
  • Carlotta Giannelli
  • Alessandra Sestini
  • Caterina Stoppato
Article

Abstract

A rotation–minimizing frame (f 1,f 2,f 3) 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 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 \(\mathbf {r}^{\prime }(\xi )=\mathcal {A}(\xi )\,\mathbf{i} \,\mathcal {A}^{*}(\xi )\) 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.

Keywords

Pythagorean–hodograph curves Rotation–minimizing frames Quaternion polynomials Rotation indicatrix 

Mathematics Subject Classification (2010)

12D05 12Y05 14H45 14H50 53A04 68U05 68U07 

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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Rida T. Farouki
    • 1
  • Graziano Gentili
    • 2
  • Carlotta Giannelli
    • 2
  • Alessandra Sestini
    • 2
  • Caterina Stoppato
    • 2
  1. 1.Department of Mechanical and Aerospace EngineeringUniversity of CaliforniaDavisUSA
  2. 2.Dipartimento di Matematica e Informatica “U. Dini”Università di FirenzeFirenzeItaly

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