Let C represent an irreducible algebraic space curve defined by the real polynomials f i ( x 1 , x 2 , x 3 ) for i = 1 , 2 . It is a recognized fact that a birational relationship invariably exists between the points on C and those on an associated irreducible plane curve, denoted as C p . In this work, we leverage this established relationship to delineate the asymptotic behavior of C by examining the asymptotes of C p . Building on this foundation, we introduce a novel and practical algorithm designed to efficiently compute the asymptotes of C , given that the asymptotes of C p have been ascertained. • Let C represent an irreducible algebraic space curve defined by the real polynomials f i ( x 1 , x 2 , x 3 ) , i = 1 , 2 . • A birational relationship always exists between the points on C and those on an associated irreducible plane curve C p . • We use this relationship to analyze the asymptotic behavior of C by examining the asymptotes of C p . • We introduce a novel algorithm designed to efficiently compute the asymptotes of C by studying the asymptotes of C p .

Given an algebraic surface implicitly defined by an irreducible polynomial, we present a method that decides whether or not this surface can be parametrized by a polynomial parametrization without base points and, in the affirmative case, we show how to compute this parametrization.

In this paper, we obtain a simple formula based on the computation of some derivatives for determining the branches and the asymptotes of curves that are defined by a parametrization. For this purpose, we use some previous results and notions presented in Blasco and Pérez-Díaz, 2014a, Blasco and Pérez-Díaz, 2014b, Blasco and Pérez-Díaz, 2015, Blasco and Pérez-Díaz, 2020. From these results, we show how the generalized asymptotes of the input curve can be easily computed and we present some applications related to the ramification index and degree of the asymptote, the infinity form and the multiplicity of the infinity points. Furthermore, we show how to construct all the families of parametric curves having some given asymptotes. We develop this method for the plane case but it can be trivially adapted for dealing with rational curves in n-dimensional space. In addition, the formulaes presented can be similarly obtained for curves defined by a parametrization not necessarily rational.

In this paper we deal with the following problem: given an algebraic plane curve C, implicitly defined, we determine its “asymptotic family”, that is, the set of algebraic curves that have the same asymptotic behavior as C.

In this paper, we present formulae for the computation of the partial degrees w.r.t. each variable of the implicit equation of a rational surface given by means of a proper parametrization. Moreover, when the parametrization is not proper we give upper bounds. These formulae generalize the results in [17] to the surface case, and they are based on the computation of the degree of the rational maps induced by the projections, onto the coordinate planes of the three dimensional space, of the input surface parametrization. In addition, using the results presented in [9] and [10], the formulae simply involve the computation of the degree of univariate polynomials directed determined from the parametrization by means of some univariate resultants and some polynomial gcds.