Representing rational curve segments and surface patches using semi-algebraic sets
Identifiers
Permanent link (URI): http://hdl.handle.net/10017/41550DOI: 10.1016/j.cagd.2019.101770
ISSN: 0167-8396
Publisher
Elsevier
Date
2019-10-01Embargo end date
2020-10-01Funders
National Natural Science Foundation of China
Agencia Estatal de Investigación
Bibliographic citation
Shen, Li-Yong, Pérez-Díaz, S., Goldman, R. & Feng, Y. 2019, “Representing rational curve segments and surface patches using semi-algebraic sets”, Computer Aided Geometric Design, vol. 74 (Oct. 2019), article 101770
Keywords
Rational curve segment
Rational surface patch
Semi-algebraic set
Implicitization
Project
Grant 61872332 (National Natural Science Foundation of China)
info:eu-repo/grantAgreement/AEI/Plan Estatal de Investigación Científica y Técnica y de Innovación 2013-2016/MTM2017-88796-P/ES/COMPUTACION SIMBOLICA: NUEVOS RETOS EN ALGEBRA Y GEOMETRIA Y SUS APLICACIONES/
Document type
info:eu-repo/semantics/article
Version
info:eu-repo/semantics/acceptedVersion
Publisher's version
https://doi.org/10.1016/j.cagd.2019.101770Rights
Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0)
© 2019 Elsevier
Access rights
info:eu-repo/semantics/openAccess
Abstract
We provide a framework for representing segments of rational planar curves or patches of rational tensor product surfaces with no singularities using semi-algebraic sets. Given a rational planar curve segment or a rational tensor product surface patch with no singularities, we find the implicit equation of the corresponding unbounded curve or surface and then construct an algebraic box defined by some additional equations and inequalities associated to the implicit equation. This algebraic box is proved to include only the given curve segment or surface patch without any extraneous parts of the unbounded curve or surface. We also explain why it is difficult to construct such an algebraic box if the curve segment or surface patch includes some singular points such as self-intersections. In this case, we show how to isolate a neighborhood of these special points from the corresponding curve segment or surface patch and to represent these special points with small curve segments or surface patches. This framework allows us to dispense with expensive approximation methods such as voxels for representing surface patches.
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