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On ▫$L^2$▫ approximation by planar Pythagorean-hodograph curvesRida T. Farouki,
Marjetka Knez,
Vito Vitrih,
Emil Žagar, 2025, original scientific article
Abstract: The ▫$L^2$▫ 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 ▫$L^2$▫ 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 ▫$L^2$▫ 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.
Keywords: ▫$L^2$▫ approximation, complex polynomial, Pythagorean-hodograph curve, Pythagorean-hodograph spline, preimage
Published in RUP: 30.05.2025; Views: 197; Downloads: 4
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