Plotting missing points and branches of real parametric curves

Article

Abstract

This paper is devoted to the study (from the theoretic and algorithmic point of view) of the existence of points and branches non-reachable by a parametric representation of a rational algebraic curve (in n-dimensional space) either over the field of complex numbers or over the field of real numbers. In particular, we generalize some of the results on missing points in (J. Symbolic Comput. 33, 863–885, 2002) to the case of space curves. Moreover, we introduce for the first time and we solve the case of missing branches. Another novelty is the emphasis on topological conditions over the curve for the existence of missing points and branches. Finally, we would like to point out that, by developing an “ad hoc” and simplified theory of valuations for the case of parametric curves, we approach in a new and unified way the analysis of the missing points and branches, and the proposal of the algorithmic solution to these problems.

Keywords

Parametric curves Normal parametrization Branches Valuation rings 

Mathematics Subject Classification (2000)

14-Q05 68-W30 

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Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Departamento de ÁlgebraUniversidad ComplutenseMadridSpain
  2. 2.Departamento de MatemáticasUniversidad de CantabriaSantanderSpain

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