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In this paper, we first summarize the algorithm presented in Blasco and Pérez-Díaz (2014) for computing the generalized asymptotes of algebraic curves implicitly defined. This algorithm is based on the computation of Puiseux series. The main and very important contribution of this paper is a new and efficient method that allows to easily compute all the generalized asymptotes of an algebraic plane curve implicitly defined by just solving a triangular system of equations. The method can be easily generalized to the case of algebraic curves implicitly defined in the n -dimensional space. [ABSTRACT FROM AUTHOR]

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. • Given a plane curve C , we determine the set of curves having the same asymptotic behavior as C. • The asymptotic behavior of a curve C is the behavior of C at the infinity. • We get the whole family of curves with the same asymptotic behavior as the given implicit curve C. • We just change the terms of degree less than d − 1 in the implicit equation of the given curve C. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of infinity branches as well as approaching curves. We present some properties which allow us to obtain an algorithm that compares the behavior of two implicitly defined algebraic plane curves at the infinity. As an important result, we prove that if two plane algebraic curves have the same asymptotic behavior, the Hausdorff distance between them is finite. [ABSTRACT FROM AUTHOR]

We develop a method for computing all the generalized asymptotes of a real plane algebraic curve implicitly defined by an irreducible polynomial . The approach is based on the notion of perfect curve introduced from the concepts and results presented in Blasco and Pérez-Díaz (2013). In addition, we study some properties concerning perfect curves and in particular, we provide a necessary and sufficient condition for a plane curve to be perfect. Finally, we show that the equivalent class of generalized asymptotes for a branch of a plane curve can be described as an affine space for a certain m. [ABSTRACT FROM AUTHOR]

In this paper, we summarize two algorithms for computing all the generalized asymptotes of a plane algebraic curve implicitly or parametrically defined. The approach is based on the notion of perfect curves introduced from the concepts and results presented in previous papers of the same authors. From these results, we derive a new and efficient method that allows to easily compute all the generalized asymptotes of an algebraic curve parametrically defined in n -dimensional space. [ABSTRACT FROM AUTHOR]