Asymptotes of space curves

In Blasco and Pérez-Díaz (2014), a method for computing generalized asymptotes of a real algebraic plane curve implicitly defined is presented. Generalized asymptotes are curves that describe the status of a branch at points with sufficiently large coordinates and thus, it is an important tool to analyze the behavior at infinity of an algebraic curve. This motivates that in this paper, we analyze and compute the generalized asymptotes of a real algebraic space curve which could be parametrically or implicitly defined. We present an algorithm that is based on the computation of the infinity branches (this concept was already introduced for plane curves in Blasco and Pérez-Díaz (2014)). In particular, we show that the computation of infinity branches in the space can be reduced to the computation of infinity branches in the plane and thus, the methods in Blasco and Pérez-Díaz (2014) can be applied

A survey on recent advances and future challenges in the computation of asymptotes

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.Ministerio de Ciencia, Innovación y Universidade

A connection between birational automorphisms of the plane and linear system of curves

In this paper, we prove that there exits a one-to-one correspondence between birational automorphisms of the plane and pairs of pencils of curves intersecting in a unique point. As a consequence, we show how to construct birational automorphisms of the plane of a certain degree d (fixed in advance) from some curves generating two linear systems of curves of degrees d and ed, where ed = d − 2 for d > 2, and ed = 1 otherwise. In addition, we also get the inverse of the birational automorphism constructed, and we show that its degree is obtained from the degree of the linear system of curves. As a special case, we show how these results can be stated to polynomial birational automorphisms of the plane

Characterizing the finiteness of the Hausdorff distance between two algebraic curves

In this paper, we present a characterization for the Hausdorff distance between two given algebraic curves in the n-dimensional space (parametrically or implicitly defined) to be finite. The characterization is related with the asymptotic behavior of the two curves and it can be easily checked. More precisely, the Hausdorff distance between two curves C and C is finite if and only if for each infinity branch of C there exists an infinity branch of C such that the terms with positive exponent in the corresponding series are the same, and reciprocally

Asymptotic behavior of an implicit algebraic plane curve

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