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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., Agencia Estatal de Investigación

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., Agencia Estatal de Investigación

In this paper, we summarize two algorithms for computing all the generalizedasymptotes of a plane algebraic curve implicitly or parametrically de-fined. The approach is based on the notion of perfect curve introducedfrom the concepts and results presented in [Blasco and Pérez-Díaz(2014)],[Blasco and Pérez-Díazz(2014-b)] and [Blasco and Pérez-Díaz(2015)]. From these results, we derive a new method that allow to easily compute horizontal and vertical asymptotes., Agencia Estatal de Investigación

We develop a method for computing all the generalized asymptotes of a real plane algebraic curve C implicitly defined by an irreducible polynomial f ( x , y ) ∈ R [ x , y ] . The approach is based on the notion of perfect curve introduced from the concepts and results presented in Blasco and Perez-Diaz (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 R m for a certain m.

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.