Improperly parametrized rational curves
Computer Aided Geometric Design
Degree, multiplicity, and inversion formulas for rational surfaces using u-resultants
Computer Aided Geometric Design
Fundamentals of computer aided geometric design
Fundamentals of computer aided geometric design
Basic algebraic geometry 1 (2nd, revised and expanded ed.)
Basic algebraic geometry 1 (2nd, revised and expanded ed.)
A rational function decomposition algorithm by near-separated polynomials
Journal of Symbolic Computation
Parametrization of algebraic curves over optimal field extensions
Journal of Symbolic Computation - Special issue: parametric algebraic curves and applications
Parametric generalized offsets to hypersurfaces
Journal of Symbolic Computation - Special issue: parametric algebraic curves and applications
The moving line ideal basis of planar rational curves
Computer Aided Geometric Design
Tracing index of rational curve parametrizations
Computer Aided Geometric Design
On multivariate rational function decomposition
Journal of Symbolic Computation - Computer algebra: Selected papers from ISSAC 2001
Partial degree formulae for rational algebraic surfaces
Proceedings of the 2005 international symposium on Symbolic and algebraic computation
On the problem of proper reparametrization for rational curves and surfaces
Computer Aided Geometric Design
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Let P be a rational affine parametrization of an algebraic surface ν, and let ϕP : K2 → v t → P(t) be the rational map induced by P. In this survey, we consider three different problems. First we deal with the problem of deciding whether ϕP is birational (i.e. whether P is proper); in case of birationality, the question of computing the inverse of the parametrization is considered. On the other side, the birationality of ϕP is also characterized by deg(ϕP) = 1. Hence the problem of analyzing the birationality is equivalent to computing deg(ϕP). The second problem considered deals with this question. More precisely, we show that deg(ϕP) can be computed by means of greatest common divisor (gcd) and univariate resultant computations. Finally, if the given parametrization P is not proper and satisfies an additional condition, we solve the problem of proper reparametrization. That is, we determine a proper rational parametrization Q(t) of ν from P such that P(t) = Q(R(t)). All the results in this survey are included in Perez-Diaz et al. (2002), Perez-Diaz and Sendra (2004) or Perez-Diaz (2006).