An isoperimetric type problem for primitive Pythagorean hodograph curves☆,☆☆
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.
References (11)
The conformal map of the hodograph plane
Computer Aided Geometric Design
(1994)Legendre–Bernstein basis transformations
J. Comput. Appl. Math.
(2000)- et al.
An isoperimetric type problem for Bézier curves of degree n
Computer Aided Geometric Design
(2010) Pythagorean-Hodograph Curves. Algebra and Geometry Inseparable
(2008)- et al.
Pythagorean hodographs
IBM Journal of Research and Development
(1990)
Cited by (3)
Identification and "reverse engineering" of Pythagorean-hodograph curves
2015, Computer Aided Geometric DesignCitation 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.
New Developments in Theory, Algorithms, and Applications for Pythagorean–Hodograph Curves
2019, Springer INdAM SeriesPythagorean hodograph curves: A survey of recent advances
2014, Journal for Geometry and Graphics
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This paper has been recommended for acceptance by R. Farouki.
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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.
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Permanent address: CIMAT, Jalisco S/N, Valenciana, Gto., C.P. 36240, Mexico.