Juan Rafael | Universidad Tecnológica de Chile (original) (raw)

Papers by Juan Rafael

Research paper thumbnail of On the Complexity of Parametrizing Curves

. Given a rational algebraic plane curve C in implicit representation weanalyze the bit complexit... more . Given a rational algebraic plane curve C in implicit representation weanalyze the bit complexity of an algorithm to describe C by parametric equations.Estimates for subalgorithms for computing the standard decomposition of singularitiesand the genus are also obtained.1. IntroductionRational parametrization of algebraic curves is one of the basic computational problems inconstructive algebraic geometry. A completely symbolic approach to this problem

Research paper thumbnail of Symbolic Parametrization of Curves

Journal of Symbolic Computation, 1991

If algebraic varieties like curves or surfaces are to be manipulated by computers, it is essentia... more If algebraic varieties like curves or surfaces are to be manipulated by computers, it is essential to be able to represent these geometric objects in an appropriate way. For some applications an implicit representation by algebraic equations is desirable, whereas for others an explicit or parametric representation is more suitable. Therefore, transformation algorithms from one representation to the other are of utmost importance.

Research paper thumbnail of Parametrization of approximate algebraic curves by lines

Theoretical Computer Science, 2004

It is well known that irreducible algebraic plane curves having a singularity of maximum multipli... more It is well known that irreducible algebraic plane curves having a singularity of maximum multiplicity are rational and can be parametrized by lines. In this paper, given a tolerance ¿ 0 and an -irreducible algebraic plane curve C of degree d having an -singularity of multiplicity d − 1, we provide an algorithm that computes a proper parametrization of a rational curve that is exactly parametrizable by lines. Furthermore, the error analysis shows that under certain initial conditions that ensures that points are projectively well deÿned, the output curve lies within the o set region of C at distance at most 2 √ 2 1=(2d) exp(2).

Research paper thumbnail of An algebraic analysis of conchoids to algebraic curves

Applicable Algebra in Engineering, Communication and Computing, 2008

We study the conchoid to an algebraic affine plane curve \({\mathcal C}\) from the perspective of... more We study the conchoid to an algebraic affine plane curve \({\mathcal C}\) from the perspective of algebraic geometry, analyzing their main algebraic properties. Beside \({\mathcal C}\) , the notion of conchoid involves a point A in the affine plane (the focus) and a non-zero field element d (the distance). We introduce the formal definition of conchoid by means of incidence diagrams. We prove that the conchoid is a 1-dimensional algebraic set having at most two irreducible components. Moreover, with the exception of circles centered at the focus A and taking d as its radius, all components of the corresponding conchoid have dimension 1. In addition, we introduce the notions of special and simple components of a conchoid. Furthermore we state that, with the exception of lines passing through A, the conchoid always has at least one simple component and that, for almost every distance, all the components of the conchoid are simple. We state that, in the reducible case, simple conchoid components are birationally equivalent to \({\mathcal C}\) , and we show how special components can be used to decide whether a given algebraic curve is the conchoid of another curve.

Research paper thumbnail of Degree formulae for offset curves

Journal of Pure and Applied Algebra, 2005

In this paper, we present three different formulae for computing the degree of the offset of a re... more In this paper, we present three different formulae for computing the degree of the offset of a real irreducible affine plane curve C given implicitly, and we see how these formulae particularize to the case of rational curves. The first formula is based on an auxiliary curve, called S, that is defined depending on a non-empty Zariski open subset of R 2 . The second formula is based on the resultant of the defining polynomial of C, and the polynomial defining generically S. The third formula expresses the offset degree by means of the degree of C and the multiplicity of intersection of C and the hodograph H to C, at their intersection points. an arbitrary algebraically closed field, of characteristic zero, we refer to . We consider C 2 as the metric affine space, induced by the inner product B(X, Y ) = X · I · Y T defined by 2 × 2 identity matrix I; note that the metric we are using is not the hermitic standard one. Now, let C be an irreducible affine plane curve over C, and let C 0 ⊂ C be the set of regular points p of C such that any non-zero normal vector to C at p is non-isotropic; i.e. the norm, in the metric affine space C 2 , of the normal vector is non-zero. Then, the offset to C, at distance d, is the Zariski closure of the constructible set A d (C) consisting of the intersection points of the circles of radius d centered at each point p ∈ C 0 and the normal line to C at p. We denote the offset to C at distance d as O d (C). We observe that, if C is given by a rational parametrization P(t), then A d (C) is essentially the set in C 2 generated by the formula P(t) ± d N(t) N(t) , where N(t) is the normal vector to C associated with the parametrization P(t). In this expression, by abuse of notation, for every non-isotropic X ∈ C 2 we write X to express any of the two numbers such that X 2 = B(X, X); if X ∈ C 2 is isotropic, then we write X = 0.

Research paper thumbnail of On the Complexity of Parametrizing Curves

. Given a rational algebraic plane curve C in implicit representation weanalyze the bit complexit... more . Given a rational algebraic plane curve C in implicit representation weanalyze the bit complexity of an algorithm to describe C by parametric equations.Estimates for subalgorithms for computing the standard decomposition of singularitiesand the genus are also obtained.1. IntroductionRational parametrization of algebraic curves is one of the basic computational problems inconstructive algebraic geometry. A completely symbolic approach to this problem

Research paper thumbnail of Symbolic Parametrization of Curves

Journal of Symbolic Computation, 1991

If algebraic varieties like curves or surfaces are to be manipulated by computers, it is essentia... more If algebraic varieties like curves or surfaces are to be manipulated by computers, it is essential to be able to represent these geometric objects in an appropriate way. For some applications an implicit representation by algebraic equations is desirable, whereas for others an explicit or parametric representation is more suitable. Therefore, transformation algorithms from one representation to the other are of utmost importance.

Research paper thumbnail of Parametrization of approximate algebraic curves by lines

Theoretical Computer Science, 2004

It is well known that irreducible algebraic plane curves having a singularity of maximum multipli... more It is well known that irreducible algebraic plane curves having a singularity of maximum multiplicity are rational and can be parametrized by lines. In this paper, given a tolerance ¿ 0 and an -irreducible algebraic plane curve C of degree d having an -singularity of multiplicity d − 1, we provide an algorithm that computes a proper parametrization of a rational curve that is exactly parametrizable by lines. Furthermore, the error analysis shows that under certain initial conditions that ensures that points are projectively well deÿned, the output curve lies within the o set region of C at distance at most 2 √ 2 1=(2d) exp(2).

Research paper thumbnail of An algebraic analysis of conchoids to algebraic curves

Applicable Algebra in Engineering, Communication and Computing, 2008

We study the conchoid to an algebraic affine plane curve \({\mathcal C}\) from the perspective of... more We study the conchoid to an algebraic affine plane curve \({\mathcal C}\) from the perspective of algebraic geometry, analyzing their main algebraic properties. Beside \({\mathcal C}\) , the notion of conchoid involves a point A in the affine plane (the focus) and a non-zero field element d (the distance). We introduce the formal definition of conchoid by means of incidence diagrams. We prove that the conchoid is a 1-dimensional algebraic set having at most two irreducible components. Moreover, with the exception of circles centered at the focus A and taking d as its radius, all components of the corresponding conchoid have dimension 1. In addition, we introduce the notions of special and simple components of a conchoid. Furthermore we state that, with the exception of lines passing through A, the conchoid always has at least one simple component and that, for almost every distance, all the components of the conchoid are simple. We state that, in the reducible case, simple conchoid components are birationally equivalent to \({\mathcal C}\) , and we show how special components can be used to decide whether a given algebraic curve is the conchoid of another curve.

Research paper thumbnail of Degree formulae for offset curves

Journal of Pure and Applied Algebra, 2005

In this paper, we present three different formulae for computing the degree of the offset of a re... more In this paper, we present three different formulae for computing the degree of the offset of a real irreducible affine plane curve C given implicitly, and we see how these formulae particularize to the case of rational curves. The first formula is based on an auxiliary curve, called S, that is defined depending on a non-empty Zariski open subset of R 2 . The second formula is based on the resultant of the defining polynomial of C, and the polynomial defining generically S. The third formula expresses the offset degree by means of the degree of C and the multiplicity of intersection of C and the hodograph H to C, at their intersection points. an arbitrary algebraically closed field, of characteristic zero, we refer to . We consider C 2 as the metric affine space, induced by the inner product B(X, Y ) = X · I · Y T defined by 2 × 2 identity matrix I; note that the metric we are using is not the hermitic standard one. Now, let C be an irreducible affine plane curve over C, and let C 0 ⊂ C be the set of regular points p of C such that any non-zero normal vector to C at p is non-isotropic; i.e. the norm, in the metric affine space C 2 , of the normal vector is non-zero. Then, the offset to C, at distance d, is the Zariski closure of the constructible set A d (C) consisting of the intersection points of the circles of radius d centered at each point p ∈ C 0 and the normal line to C at p. We denote the offset to C at distance d as O d (C). We observe that, if C is given by a rational parametrization P(t), then A d (C) is essentially the set in C 2 generated by the formula P(t) ± d N(t) N(t) , where N(t) is the normal vector to C associated with the parametrization P(t). In this expression, by abuse of notation, for every non-isotropic X ∈ C 2 we write X to express any of the two numbers such that X 2 = B(X, X); if X ∈ C 2 is isotropic, then we write X = 0.