An isoperimetric type problem for primitive Pythagorean hodograph curves

doi:10.1016/j.cagd.2012.06.002

Computer Aided Geometric Design

Volume 29, Issue 8, November 2012, Pages 626-647

https://doi.org/10.1016/j.cagd.2012.06.002 Get rights and content

Abstract

An isoperimetric type problem for primitive Pythagorean hodograph curves is studied. We show how to compute, for each possible degree, the Pythagorean hodograph curve of a given perimeter enclosing the greatest area. We also discuss the existence and construction of smooth solutions, obtaining a relationship with an interesting sequence of Appell polynomials.

Highlights

► An isoperimetric type problem for primitive Pythagorean hodograph curves is studied. ► We show how to compute, for each possible degree, the Pythagorean hodograph curve of a given perimeter enclosing the greatest area. ► The existence and construction of smooth solutions is discussed, obtaining a relationship with an interesting sequence of Appell polynomials.

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There are more references available in the full text version of this article.

Cited by (3)

Identification and "reverse engineering" of Pythagorean-hodograph curves
2015, Computer Aided Geometric Design
Citation Excerpt :
Specifically, planar and spatial Pythagorean hodographs are constructed by quadratic mappings of complex and quaternion polynomials, respectively, and the coefficients of these pre-image polynomials are required for the exact determination of various PH curve properties. Many methods for the construction of planar and spatial PH curves are available (Farouki et al., 2008, 2001; Farouki and Neff, 1995; Habib and Sakai, 2007; Huard et al., 2014; Jaklic et al., 2012; Jüttler and Mäurer, 1999; Jüttler, 2001; Klar and Valasek, 2011; Monterde and Ongay, 2012; Walton and Meek, 1998, 2002). The goal of this study is to facilitate their importation into commercial CAD systems through existing CAD data formats, by developing algorithms that (i) identify whether or not specified Bézier/B-spline data define a PH curve; and (ii) if so, reconstruct its “internal structure” variables.
Methods are developed to identify whether or not a given polynomial curve, specified by Bézier control points, is a Pythagorean-hodograph (PH) curve — and, if so, to reconstruct the internal algebraic structure that allows one to exploit the advantageous properties of PH curves. Two approaches to identification of PH curves are proposed. The first is based on the satisfaction of a system of algebraic constraints by the control-polygon legs, and the second uses the fact that numerical quadrature rules that are exact for polynomials of a certain maximum degree generate arc length estimates for PH curves exhibiting a sharp saturation as the number of sample points is increased. These methods are equally applicable to planar and spatial PH curves, and are fully elaborated for cubic and quintic PH curves. The reverse engineering problem involves computing the complex or quaternion coefficients of the pre-image polynomials generating planar or spatial Pythagorean hodographs, respectively, from prescribed Bézier control points. In the planar case, a simple closed-form solution is possible, but for spatial PH curves the reverse engineering problem is much more involved.
New Developments in Theory, Algorithms, and Applications for Pythagorean–Hodograph Curves
2019, Springer INdAM Series
Pythagorean hodograph curves: A survey of recent advances
2014, Journal for Geometry and Graphics

^☆: This paper has been recommended for acceptance by R. Farouki.

^☆☆: This work is partially supported by grant MTM2009-08933 from the Spanish Ministry of Science and Innovation, and by CONACYT (Mexico), project 106 923. The second author also wishes to thank the Departament de Geometria i Topologia of the Universitat de València, for its kind hospitality while in a sabbatical stay, also supported by a CONACYT grant.

¹: Permanent address: CIMAT, Jalisco S/N, Valenciana, Gto., C.P. 36240, Mexico.

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An isoperimetric type problem for primitive Pythagorean hodograph curves☆,☆☆

Abstract

Highlights

Computer Aided Geometric Design

J. Comput. Appl. Math.

Computer Aided Geometric Design

Pythagorean-Hodograph Curves. Algebra and Geometry Inseparable

Pythagorean hodographs

IBM Journal of Research and Development