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Spatial C2 closed loops of prescribed arc length defined by Pythagorean-hodograph curves

Author

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  • Farouki, Rida T.
  • Knez, Marjeta
  • Vitrih, Vito
  • Žagar, Emil

Abstract

We investigate the problem of constructing spatial C2 closed loops from a single polynomial curve segment r(t), t∈[0,1] with a prescribed arc length S and continuity of the Frenet frame and curvature at the juncture point r(1)=r(0). Adopting canonical coordinates to fix the initial/final point and tangent, a closed-form solution for a two-parameter family of interpolants to the given data can be constructed in terms of degree 7 Pythagorean-hodograph (PH) space curves, and continuity of the torsion is also obtained when one of the parameters is set to zero. The geometrical properties of these closed-loop PH curves are elucidated, and certain symmetry properties and degenerate cases are identified. The two-parameter family of closed-loop​ C2 PH curves is also used to construct certain swept surfaces and tubular surfaces, and a selection of computed examples is included to illustrate the methodology.

Suggested Citation

  • 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).
    • Handle: RePEc:eee:apmaco:v:391:y:2021:i:c:s009630032030607x
    DOI: 10.1016/j.amc.2020.125653
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