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On L2 approximation by planar Pythagorean-hodograph curves

Author

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

Abstract

The L2 approximation of planar curves by Pythagorean-hodograph (PH) polynomial curves is addressed, based on the distance defined by a metric for planar curves represented as complex-valued functions of a real parameter. Because of the nonlinear nature of polynomial PH curves, constructing L2 approximants involves solving a nonlinear optimization problem. However, a simplified method that requires only the solution of a linear system may be developed by formulating the L2 approximation in the preimage space. The extension of the methodology to approximation by PH B-spline curves is also addressed, and several examples are provided to illustrate its implementation and potential.

Suggested Citation

  • Farouki, Rida T. & Knez, Marjeta & Vitrih, Vito & Žagar, Emil, 2025. "On L2 approximation by planar Pythagorean-hodograph curves," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 233(C), pages 296-310.
  • Handle: RePEc:eee:matcom:v:233:y:2025:i:c:p:296-310
    DOI: 10.1016/j.matcom.2025.02.001
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    Keywords

    L2 approximation; Complex polynomial; Pythagorean-hodograph curve; Pythagorean-hodograph spline; Preimage;
    All these keywords.

    JEL classification:

    • L2 - Industrial Organization - - Firm Objectives, Organization, and Behavior

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