Improperly parametrized rational curves
Computer Aided Geometric Design
Geometric and solid modeling: an introduction
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Algorithmic algebra
Fundamentals of computer aided geometric design
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Basic algebraic geometry 1 (2nd, revised and expanded ed.)
Basic algebraic geometry 1 (2nd, revised and expanded ed.)
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Journal of Symbolic Computation - Special issue: parametric algebraic curves and applications
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Journal of Symbolic Computation - Special issue: parametric algebraic curves and applications
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ISSAC '97 Proceedings of the 1997 international symposium on Symbolic and algebraic computation
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ISSAC '99 Proceedings of the 1999 international symposium on Symbolic and algebraic computation
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Computer Aided Geometric Design
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Proceedings of the 2006 international symposium on Symbolic and algebraic computation
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Journal of Symbolic Computation
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Proceedings of the 2007 international workshop on Symbolic-numeric computation
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Journal of Symbolic Computation
On the different shapes arising in a family of plane rational curves depending on a parameter
Computer Aided Geometric Design
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Computer Aided Geometric Design
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In this paper we present a theoretical and algorithmic analysis on the normality of rational parameterizations of algebraic plane curves over arbitrary fields of characteristic zero. If the field is algebraically closed we give an algorithm to decide whether a parametrization is proper and, if not, a normal parametrization is computed. If the field is not algebraically closed the problem is more complicated, and a degenerated situation may appear. We classify the degenerations in strong and weak degenerations, and an algorithm to decide this phenomenon is derived. Furthermore, we prove that if the parametrization is strongly degenerated then the curve cannot be normally parametrized, but weak degenerations can be resolved, and an algorithm to reparametrize the input weakly degenerated parametrization into a non-degenerated one is given. In addition, we show how these results can be applied and improved to the case of real rational curves. In this case, we present an algorithm that decides whether a given real parametrization can be normally parametrized over the reals, and that computes such a parametrization if it exists.