Title: Piecewise rational approximation of square-root parameterizable curves using the Weierstrass form
Authors: Bizzarri, Michal
Lávička, Miroslav
Vršek, Jan
Citation: BIZZARRI, M., LÁVIČKA, M., VRŠEK, J. Piecewise rational approximation of square-root parameterizable curves using the Weierstrass form. Computer aided geometric design, 2017, roč. 56, č. August, s. 52-66. ISSN 0167-8396.
Issue Date: 2017
Publisher: Elsevier
Document type: preprint
URI: http://hdl.handle.net/11025/29225
ISSN: 0167-8396
Keywords: parametrizace odmocniny;hypereliptické křivky;Weierstrassova forma;topologický graf;racionální aproximace
Keywords in different language: Square-root parameterization;hyperelliptic curves;Weierstrass form;topological graph;rational approximation
Abstract in different language: In this paper we study situations when non-rational parameterizations of planar or space curves as results of certain geometric operations or constructions are obtained, in general. We focus especially on such cases in which one can identify a rational mapping which is a double cover of a rational curve. Hence, we deal with rational, elliptic or hyperelliptic curves that are birational to plane curves in the Weierstrass form and thus they are square-root parameterizable. We design a simple algorithm for computing an approximate (piecewise) rational parametrization using topological graphs of the Weierstrass curves. Predictable shapes reflecting a number of real roots of a univariate polynomial and a possibility to approximate easily the branches separately play a crucial role in the approximation algorithm. Our goal is not to give a comprehensive list of all such operations but to present at least selected interesting cases originated in geometric modelling and to show a unifying feature of the formulated method. We demonstrate our algorithm on a number of examples.
Rights: Plný text není přístupný.
© Elsevier
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