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Parametric curves with Pythagorean binormal

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Abstract

In this paper, a class of rational spatial curves that have a rational binormal is introduced. Such curves (called PB curves) play an important role in the derivation of rational rotation-minimizing osculating frames. The PB curve construction proposed is based upon the dual curve representation and the Euler-Rodrigues frame obtained from quaternion polynomials. The construction significantly simplifies if the curve is a polynomial one. Further, polynomial PB curves of the degree ≥ 7 and rational PB curves of the degree ≥ 6 that possess rational rotation-minimizing osculating frames are derived, and it is shown that no lower degree curves, constructed from quadratic quaternion polynomials, with such a property exist.

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Authors and Affiliations

  1. FMF and IMFM, University of Ljubljana, Jadranska 19, 1000, Ljubljana, Slovenia

    Jernej Kozak & Marjeta Krajnc

  2. IAM and FAMNIT, University of Primorska, Muzejski trg 2, 6000, Koper, Slovenia

    Vito Vitrih

Authors
  1. Jernej Kozak
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  2. Marjeta Krajnc
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  3. Vito Vitrih
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Corresponding author

Correspondence to Marjeta Krajnc.

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Communicated by: Rida T. Farouki

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Kozak, J., Krajnc, M. & Vitrih, V. Parametric curves with Pythagorean binormal. Adv Comput Math 41, 813–832 (2015). https://doi.org/10.1007/s10444-014-9387-7

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  • DOI: https://doi.org/10.1007/s10444-014-9387-7

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