arXiv subjects

Angel Blasco

Publications and source records attributed to Angel Blasco.

5 recordsLinked to original sources

arXiv2017

The limit point and the T--function

Let ${\cal P}(t)\in {\Bbb K}(t)^{n}$ be a rational parametrization of an algebraic space curve $\cal C$. In this paper, we introduce the notion of limit point, $P_L$, of the given parametrization $\mathcal{P}(t)$, and some remarkable properties of $P_L$ are obtained. In addition, we generalize the results in \cite{MyB-2017} concerning the T--function, $T(s)$, which is defined by means of a univariate resultant. More precisely, independently on whether the limit point is regular or not, we show that $T(s)=\prod_{i=1}^n H_{P_i}(s)^{m_i-1}$. The polynomials $H_{P_i}(s),\,i=1,\ldots,n$ are the fibre functions, and its roots are the fibre of the ordinary singularities $P_i\in {\cal C}$ of multiplicity $m_i,\,i=1,\ldots,n$. Thus, a complete classification of the singularities of a given space curve, via the factorization of a resultant, is obtained.

math.AG↗

arXiv2017

Resultants and Singularities of Parametric Curves

Let ${\cal C}$ be an algebraic space curve defined parametrically by ${\cal P}(t)\in {\Bbb K}(t)^{n},\,n\geq 2$. In this paper, we introduce a polynomial, the T--function, $T(s)$, which is defined by means of a univariate resultant constructed from ${\cal P}(t)$. We show that $T(s)=\prod_{i=1}^n H_{P_i}(s)^{m_i-1}$, where $H_{P_i}(s),\,i=1,\ldots,n$ are polynomials (called the fibre functions) whose roots are the fibre of the ordinary singularities $P_i\in {\cal C}$ of multiplicity $m_i,\,i=1,\ldots,n$. Thus, a complete classification of the singularities of a given space curve, via the factorization of a resultant, is obtained.

math.AG↗

arXiv2014

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 $\cal C$ and $\overline{\cal C}$ is finite if and only if for each infinity branch of $\cal C$ there exists an infinity branch of $\overline{\cal C}$ such that the terms with positive exponent in the corresponding series are the same, and reciprocally.

math.AG↗

arXiv2014

Asymptotes of Space Curves

In this paper, we generalize the results presented in [5] for the case of real algebraic space curves. More precisely, given an algebraic space curve C (parametrically or implicitly defined), we show how to compute the generalized asymptotes. The approach is based on the notion of perfect curve introduced from the concepts and results presented in [4].

math.AG↗

arXiv2013

Asymptotes and Perfect Curves

We develop a method for computing all the {\it generalized asymptotes} of a real plane algebraic curve $\cal C$ over $\Bbb C$ implicitly defined by an irreducible polynomial $f(x,y)\in {\Bbb R}[x,y]$. The approach is based on the notion of perfect curve introduced from the concepts and results presented in Blasco, A., Pérez-D\'ıaz, S. (2013). {\it Asymptotic Behavior of an Implicit Algebraic Plane Curve}. arxiv.org/abs/1302.2522v2.

math.AG↗