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Three paths to rational curves with rational arc length

Author

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  • Schröcker, Hans-Peter
  • Šír, Zbyněk

Abstract

We solve the so far open problem of constructing all spatial rational curves with rational arc length functions. More precisely, we present three different methods for this construction. The first method adapts a recent approach of (Kalkan et al. 2022) to rational PH curves and requires solving a modestly sized system of linear equations. The second constructs the curve by imposing zero-residue conditions, thus extending ideas of previous papers by (Farouki and Sakkalis 2019) and the authors themselves (Schröcker and Šír 2023). The third method generalizes the dual approach of (Pottmann 1995) from planar to spatial curves. The three methods share the same quaternion based representation in which not only the PH curve but also its arc length function are compactly expressed. We also present a new proof based on the quaternion polynomial factorization theory of the well known characterization of the Pythagorean quadruples.

Suggested Citation

  • Schröcker, Hans-Peter & Šír, Zbyněk, 2024. "Three paths to rational curves with rational arc length," Applied Mathematics and Computation, Elsevier, vol. 478(C).
    • Handle: RePEc:eee:apmaco:v:478:y:2024:i:c:s0096300324003035
    DOI: 10.1016/j.amc.2024.128842
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    References listed on IDEAS

    as
    1. Knez, Marjeta & Pelosi, Francesca & Sampoli, Maria Lucia, 2022. "Construction of G2 planar Hermite interpolants with prescribed arc lengths," Applied Mathematics and Computation, Elsevier, vol. 426(C).
    2. Farouki, Rida T. & Knez, Marjeta & Vitrih, Vito & Žagar, Emil, 2021. "Spatial C2 closed loops of prescribed arc length defined by Pythagorean-hodograph curves," Applied Mathematics and Computation, Elsevier, vol. 391(C).
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