Laureano González-Vega

CUNEF · Quantitative Methods

The problem of computing the topology of curves has received special attention from both Computer Aided Geometric Design and Symbolic Computation. It is well known that the general position condition simplifies the computation of the topology of a real algebraic plane curve defined implicitly since, under this assumption, singular points can be pre...

In this paper, using Descartes law of sign, we provide elementary proof of results on the number of real eigenvalues of real matrices of which certain properties on the signs of their principal minors are known. More precisely, we analyze P, N, Q, R, PN and $QR$ matrices as well as their variants "almost", "weak" and "sub-zero" matrices.

The problem of detecting when two moving ellipses or ellipsoids overlap is of interest to robotics, CAD/CAM, computer animation, etc., where ellipses and ellipsoids are often used for modelling (and/or enclosing) the shape of the objects under consideration. By analysing symbolically the sign of the real roots of the characteristic polynomial of th...

In this paper, for certain type of structured {0,1,−1}–matrices, we give a complete description of the inner Bohemian inverses over any population containing the set {0,1,−1}. In addition, when the population is exactly {0,1,−1}, we provide explicit formulas for the number of inner Bohemian inverses of these type of matrices.

A new determinantal representation for the implicit equation of offsets to conics and quadrics is derived. It is simple, free of extraneous components and provides a very compact expanded form, these representations being very useful when dealing with geometric queries about offsets such as point positioning or solving intersection purposes. It is...

This article introduces several efficient and easy-to-use tools to analyze the intersection curve between two quadrics, on the basis of the study of its projection on a plane (the so-called cutcurve) to perform the corresponding lifting correctly. This approach is based on an efficient way of determining the topology of the cutcurve through only so...

A set of matrices with entries from a fixed finite population P is called “Bohemian”. The mnemonic comes from BOunded HEight Matrix of Integers, BOHEMI, and although the population P need not be solely made up of integers, it frequently is. In this paper we look at Bohemians, specifically those with population {−1,0,+1} and sometimes other populati...

The Maple intersectplot command plots the intersection curve in three-dimensional space between a pair of two-dimensional surfaces. We will present the implementation in Maple of a new algorithm computing the intersection curve between two quadrics in 3D that improves the results produced by the intersectplot command.

We look at Bohemians, specifically those with population $\{-1, 0, {+1}\}$ and sometimes $\{0,1,i,-1,-i\}$. More, we specialize the matrices to be upper Hessenberg Bohemian. From there, focusing on only those matrices whose characteristic polynomials have maximal height allows us to explicitly identify these polynomials and give useful bounds on th...

This paper presents the first implementation in GeoGebra of an algorithm computing the intersection curve of two quadrics. This approach is based on computing the projection of the intersection curve, also known as cutcurve, determining its singularities and structure and lifting to 3D this plane curve. The considered problem can be used to show so...

This paper is devoted to presenting a method to determine the intersection of two quadrics based on the detailed analysis of its projection in the plane (the so called cutcurve) allowing to perform the corresponding lifting correctly. This approach is based on a new computational characterisation of the singular points of the curve and on how this...

A new determinantal presentation of the implicit equation for offsets to non degenerate conics and quadrics is introduced which is specially well suited for intersection purposes.

We show how to construct linearizations of matrix polynomials za(z)d0+c0, a(z)b(z), a(z)+b(z) (when deg(b(z))<deg(a(z))), and za(z)d0b(z)+c0 from linearizations of the component parts, a(z) and b(z). This allows the extension to matrix polynomials of a new companion matrix construction.

We look at Bohemian matrices, specifically those with entries from $\{-1, 0, {+1}\}$. More, we specialize the matrices to be upper Hessenberg, with subdiagonal entries $\pm1$. Many properties remain after these specializations, some of which surprised us. We find two recursive formulae for the characteristic polynomials of upper Hessenberg matrices...

We look at Bohemian matrices, specifically those with entries from $\{-1, 0, {+1}\}$. More, we specialize the matrices to be upper Hessenberg, with subdiagonal entries $1$. Even more, we consider Toeplitz matrices of this kind. Many properties remain after these specializations, some of which surprised us. Focusing on only those matrices whose char...

We give a general theory of generalised inverses and we explain the link with the theory of finitely generated projective modules. All the paper is written in constrctive mathematics in Bishop style. So all results do have a clear algorithmic content. We give also a complexity analysis of the algorihms corresponding to the main theorems. Here is a...

The problem of detecting when two moving ellipsoids overlap is of interest to robotics, CAD/CAM, computer animation, etc. By analysing symbolically the sign of the real roots of the characteristic polynomial of the pencil defined by two ellipsoids \(\mathcal {A}\) and \(\mathcal {B}\) we use and analyse the new closed formulae introduced in [9] cha...

We show how to construct linearizations of matrix polynomials $z\mathbf{a}(z)\mathbf{d}_0 + \mathbf{c}_0$, $\mathbf{a}(z)\mathbf{b}(z)$, $\mathbf{a}(z) + \mathbf{b}(z)$ (when $\mathrm{deg}\left(\mathbf{b}(z)\right) < \mathrm{deg}\left(\mathbf{a}(z)\right)$), and $z\mathbf{a}(z)\mathbf{d}_0\mathbf{b}(z) + \mathbf{c_0}$ from linearizations of the com...

The fact that a real univariate polynomial misses some real roots is usually overcame by considering complex roots, but the price to pay for, is a complete lost of the sign structure that a set of real roots is endowed with (mutual position on the line, signs of the derivatives, etc...). In this paper we present real substitutes for these missing r...

A new algebraic framework is introduced for computing the topology of the offset Cδ at distance δ to a rational plane curve C defined by a parameterization (x(t),y(t)). The focus is on computing the topology of Cδ by analyzing the image of the parameterization of Cδ which involves square roots. This framework is mainly intended to deal with curves...

Global and local positioning systems (LPS) make use of nonlinear equations systems to calculate coordinates of unknown points. There exist several methods, such as Sturmfels’ resultant, Groebner bases and least squares, for dealing with this kind of equations. We introduce two methods for solving this problem with the aid of symbolic techniques rel...

A new approach is presented for computing the medial axis of a planar, closed and bounded domain whose boundary consists of nitely many segments and conic arcs. The new method is topologically correct (no components are missed) and geometrically exact (each component is represented exactly).

The aim of this paper is to introduce the construction of the Bézout matrix of two univariate polynomials given by values in the Hermite interpolation basis, namely the confluent Bézout matrix. Moreover, if such polynomials have exactly one common simple zero, we describe how to compute it from the null space of the confluent Bézout matrix.

We present in this paper a canonical form for the elements in the ring of continuous piecewise polynomial functions. This new representation is based on the use of a particular class of functions $$\{C_i(P):P\in\Q[x],i=0,\ldots,\deg(P)\}$$ defined by $$C_i(P)(x)= \left\{ \begin{array}{cll}0 & \mbox{ if } & x \leq \alpha \\ P(x) & \mbox{ if } & x \g...

Curve arrangement studying is a subject of great interest in Computational Geometry and CAGD. In our paper, a new method for computing the topology of an arrangement of algebraic plane curves, defined by implicit and parametric equations, is presented. The polynomials appearing in the equations are given in the Lagrange basis, with respect to a sui...

This work has three main parts: the bisectors of curves in the plane, the bisectors of surfaces in ℝ3, and the Voronoï diagram of a finite family of parallel half-lines in ℝ3. These subjects are closely related, and have applications in CAD/CAGD and Computational Geometry.We present a new approach to determine, using Cramer’s rule and suitable elim...

The bisectors are geometric constructions with different applications in tool path generation, motion planning, NC-milling, etc. We present a new approach to determine an algebraic representation (parameterization or implicit equation) of the bisector surface of two given low degree parametric surfaces. The method uses the so-called generalized Cra...

This paper is devoted to introducing a new approach for computing the topology of a real algebraic plane curve presented either parametrically or defined by its implicit equation when the corresponding polynomials which describe the curve are known only “by values”. This approach is based on the replacement of the usual algebraic manipulation of th...

The availability of the implicit equation of a plane curve or of a 3D surface can be very useful in order to solve many geometric problems involving the considered curve or surface: for example, when dealing with the point position problem or answering intersection questions. On the other hand, it is well known that in most cases, even for moderate...

Bisectors are geometric constructions with applications in Tool path genera- tion, Motion planning, NC-milling, etc. For two given low degree parametric surfaces, it will be presented a new approach to determine an algebraic representation of their bisector by using the so-called generalized Cramer rules. The new introduced approach allows to easil...

TutorMates is a new educational software designed to combine symbolic and geometric tools to teach and learn mathematics at Secondary Education level. Its main technical ingredients are: a Computer Algebra system (Maxima) plus a dynamic geometry and graphical tool (GeoGebra), and an intermediary layer that connects the user interface and the compu...

We introduce a unifying formulation of a number of related problems which can all be solved using a contour integral formula. Each of these problems requires finding a non-trivial linear combination of possibly some of the values of a function f, and possibly some of its derivatives, at a number of data points. This linear combination is required t...

We propose a novel approach for the approximate parameterization of an implic- itly defined curve in the plane by polynomial parametric spline curves. The method generates the parameterization of the curve (which may consist of several open and closed branches) without using any a priori information about its topology.

This special issue contains a selection of papers that were presented at the 24th European Workshop on Computational Geometry (EuroCG'08) held in Nancy, France, on March 18-20, 2008. This workshop is an annual event dedicated to promoting research in Computational Geometry and related fields, within Europe and beyond. Almost 140 participants attend...

The distance of closest approach of two separated ellipses and ellipsoids is the distance among their centres when they are externally tangent after moving them through the line joining their centres. This is a critical problem when modeling and simulating systems of anisometric particles such as liquid crystals because that distance in the case of...

From the rational functions defining a rational plane curve it is possible to construct two bivariate polynomials that can be seen as univariate polynomials in the parameter value. In this paper several relevant properties of the subresultant sequence of these two polynomials are introduced which are used to solve simultaneously the implicitization...

Closed form solutions for transforming 3D Cartesian to geodetic coordinates reduce the problem to finding the real solutions of the fourth degree latitude equation or variations of it. By using symbolic tools (Sturm–Habicht coefficients and subresultants mainly) we study the methods (and polynomials) proposed by Vermeille and Borkowski to solve thi...

The support function of a free-form-surface is closely related to the implicit equation of the dual surface, and the process of computing both the dual surface and the support function can be seen as dual implicitization. The support function can be used to parameterize a surface by its inverse Gauss map. This map makes it relatively simple to stud...

The main purpose of this paper is to analyze new determinantal expressions which define the subresultant sequence of two polynomials, when the coefficients of such polynomials depend on parameters.

We review different group based algorithms for matrix multiplication and discuss the relations between the combinatorial properties of the used group and the complexity of these algorithms. We introduce a variant of an algorithm based on the ideas exposed in [H. Cohn, R. Kleinberg and B. Szegedy, in: FOCS 2005: 46th Annual IEEE Symposium on Foundat...

We present a new type of oriented bounding surfaces, which is particularly well suited for shortest distance computations. The bounding surfaces are obtained by considering surfaces whose support functions are restrictions of quadratic polynomials to the unit sphere. We show that the common normals of two surfaces of this type - and hence their sho...

We present a new type of oriented bounding surfaces, which is particularly well suited for shortest distance computations. The bounding surfaces are obtained by considering surfaces whose support functions are restrictions of quadratic polynomials to the unit sphere. We show that the common normals of two surfaces of this type - and hence their sho...

The aim of this paper is to investigate the possibility of solving a linear differential equation of degree n by means of differential equations of degree less than or equal to a fixed d, 1@?d

We discuss rational parameterizations of surfaces whose support functions are rational functions of the coordinates specifying the normal vector and of a given non-degenerate quadratic form. The class of these surfaces is closed under offsetting. It comprises surfaces with rational support functions and non-developable quadric surfaces, and it is a...

We introduce a new algebraic approach dealing with the problem of computing the topology of an arrangement of a finite set of real algebraic plane curves presented implicitly. The main achievement of the presented method is a complete avoidance of irrational numbers that appear when using the sweeping method in the classical way for solving the pro...

This paper is devoted to improve the efficiency of the algorithm introduced in [A. Eigenwillig, L. Kettner, E. Schömer, N. Wolpert, Exact, efficient and complete arrangement computation for cubic curves, Computational Geometry 35 (2006) 36–73] for analyzing the topology of an arrangement of real algebraic plane curves by using deeper the well-known...

Continuous extensions are now routinely provided by many IVP solvers, for graphical output, error control, or event location. Recent developments suggest that a uniform, stable and convenient interpolant may be provided directly by value and derivative data (Hermite data), because a new companion matrix for such data allows stable, robust and conve...

We consider surfaces whose support function is obtained by restricting a quadratic polynomial to the unit sphere. If such a surface is subject to a rigid body motion, then the Gauss image of the characteristic curves is shown to be a spherical quartic curve, making them accessible to exact geometric computation. In particular we analyze the case of...

A new seminumerical algorithm for computing the intersection curve between a plane and the offset of a parametric surface is presented. The corresponding implementation and the performed experimentation are also reported.

This paper shows how the efficiency of the current methodologies applied to the surface-to-surface intersection problem can be improved by combining an algebraic/symbolic framework with efficient and robust numerical techniques. The algebraic/symbolic framework is used to translate the computation of resultants, subresultants, discriminants, etc. t...

We analyze how to compute in an efficient way the topology of an arrangement of quartic curves. We suggest a sweeping method that generalizes the one presented by Eigenwillig et al. for cubics. The proposed method avoids working with the roots of the involved resultants (most likely algebraic numbers) in order to give an exact and complete answer....

We briefly present and analyze, from a geometric viewpoint, strategies for designing algorithms to factor bivariate approximate polynomials in C[x; y]. Given a composite polynomial, stably square-free, satisfying a genericity hypothesis, ...

Using a new formulation of the Bézout matrix, we construct bivariate matrix polynomials expressed in a tensor-product Lagrange basis. We use these matrix polynomials to solve common tasks in computer-aided geometric design. For example, we show that these bivariate polynomials can serve as stable and efficient implicit representations of plane curv...

Real implicitization of parametric curves has important applications in computer aided geometric design. Implicitization of parametric curves by resultant computations may lead to super uous isolated points. Hence, an exact implicit description should consist of equations and further conditions excluding these geometric extraneous components. Altho...

We present a collection of methods and tools for computing the topology of real algebraic plane curves de .ned by bivariate polynomial equations that are known at certain values or easy to evaluate, but whose explicit description is not available.The principal techniques used are the reduction of the computation of the real roots of the discriminan...

Using a new formulation of the Bézout matrix, we construct bivariate matrix polynomials expressed in a tensor-product Lagrange basis. We use these matrix polynomials to solve common tasks in computer-aided geometric design. For ex-ample, we show that these bivariate polynomials can serve as stable and efficient implicit representations of plane cur...

We propose a hybrid numerical-symbolic algorithm determining the intersection curve between two triangular patches when they meet transversally for the three possible cases, depending on how the patches are presented, parametrically or implicitly.

This paper presents a new algorithm for computing the intersection of a rational revolution surface or a canal surface, given in parametric or implicit form, and another surface given in parametric form. The problem is reduced to finding the zero set of a bivariate equation which represents the parameter values of the intersection curve, as a subse...

D'une part, nous développons la théorie générale des inverses généralisés de matrices en la mettant en rapport avec la théorie constructive des modules projectifs de type fini. D'autre part nous précisons certains aspects de cette théorie liés au calcul formel et à l'analyse numérique matricielle. Nous démontrons en particulier qu'on peut tester si...

By using several tools coming from Real Algebraic Geometry, Computer Algebra and Projective Geometry (Sturm–Habicht sequences and the classification of pencils of conics in P2(R)), a new approach for characterizing the ten relative positions of two ellipses is introduced.Each relative position is exclusively characterized by a set of equalities and...

Barnett’s method through Bezoutians is a purely linear algebra method allowing to compute the degree of the greatest common divisor of several univariate polynomials in a very compact way. Two different uses of this method in computer algebra are introduced here. Firstly, we describe an algorithm for parameterizing the greatest common divisor of se...

In this paper we generalize a method to analyze inhomogeneous polynomial systems containing parameters. In particular, the Hilbert function is used as a tool to check that the specialization of a “generic” Gröbner basis of the parametric polynomial system (computed in a polynomial ring having both parameters and unknowns as variables) is a Gröbner...

Following Mulmuley's lemma, this paper presents a generalization of the Moore--Penrose inverse for a matrix over an arbitrary field. This generalization yields a way to uniformly solve linear systems of equations which depend on some parameters.

In this paper a new algorithm for computing the intersection of two rational ruled surfaces, given in parametric/parametric or implicit/parametric form, is presented. This problem can be considered as a quantifier elimination problem over the reals with an additional geometric flavor which is one of the central themes in V. Weispfenning research. A...

This paper is devoted to characterizing the geometric extraneous components which may arise when trying to use the implicit description of a parametric curve or surface. This is especially critical for the (parametric) surface-to-surface intersection problem, since these components are a source of numerical problems or even incorrect points in the...

This article is devoted to presenting new expressions for Subresultant Polynomials, written in terms of some minors of matrices different from the Sylvester matrix. Moreover, via these expressions, we provide new proofs for formulas which associate the Subresultant polynomials and the roots of the two polynomials. By one hand, we present a new proo...

The main purpose of this article is to present algorithms to parameterize the degree of the greatest common divisor of two polynomials with parametric coefficients: these algorithms are based on the fact that the principal minors of the Bezout matrices provide the principal subresultant sequence. When coefficients depend on parameters, these algori...

Authors, we can assume, start writing a book with pleasure. As their project progresses, their basic mood might turn to hope (of soon getting to the end of an overwhelming project). Conversely, readers, we assume, to start browsing a book with hope (of learning something interesting) but only in some special cases do they continue reading it with p...

We show how to adapt algebraic techniques to deal with some problems, such as implicitization or parameterization, in computer aided geometric design that involve floating-point real numbers. In addition, applications to the offsetting and blending problems are also considered.

This chapter is devoted to showing how to solve the intersection problem for two algebraic surfaces presented implicitly by using algebraic techniques for manipulating systems of polynomial equations, the numerical solution of first order systems of differential equations and the properties of scalar and gradient vector fields.

Inspired by classical results in algebraic geometry, we study the continuity with respect to the coefficients, of the zero set of a system of complex homogeneous polynomials with a given pattern and when the Hilbert polynomial of the generated ideal is fixed. In this work we prove topological properties of some classifying spaces, e.g. the space of...

We present a collection of methods and tools for computing the topology of real curves defined by bivariate polynomial equations. The principal used techniques are the reduction of the computation of the real roots of the discriminant to a sparse generalized eigenvalue problem and the use of the structure of the nullspace of the classical Bezoutian...

Se describen los trabajos realizados, para la titulación de Matemáticas, dentro de los proyectos "Tuning Educational Estructures in Europe" y "Proyecto-Piloto CRUE". Asimismo se analiza el nuevo marco normativo que conducirá a la integración del sistema universitario español en el Espacio Europeo de la Educación Superior y la futura elaboración del...

This paper is devoted to present a new algorithm computing in a very efficient way the topology of a real algebraic plane curve defined implicitly. This algorithm proceeds in a seminumerical way by performing a symbolic preprocessing which allows later to accomplish the numerical computations in a very accurate way.

This paper is devoted to show, first, how to easily determine, when it exists, a non-trivial element in the centre of the Galois group of an irreducible polynomial in ℤ[x] and, second, how to deal in an efficient way with solvability by radicals when a non-trivial element in the centre of the Galois group of the considered polynomial is available.

This article provides a new presentation of Barnett’s theorems giving the degree (resp. coefficients) of the greatest common divisor of several univariate polynomials with coefficients in an integral domain by means of the rank (resp. linear dependencies of the columns) of several Bezout-like matrices. This new presentation uses Bezout or hybrid Be...

This paper is devoted to presenting, first, a family of formulas extending to the multivariate case the classical Newton (or Newton-Girard) identities relating the coefficients of a univariate polynomial equation with its roots through the Newton sums and, secondly, the generating functions associate to the newly introduced Newton sums of the multi...

We consider a planar rational cubic Bézier curve C(u), with u∈[0,1] and we implicitize C(u) in the form of a cubic Bernstein-Bézier triangular patch f i.e. we find a non-zero function f:ℝ 2 →ℝ, defined in terms of barycentric coordinates, for wich f(C(u))=0 for all u. We make the assumption that the end tangents (C ' (0)andC ' (1)) are parallel.

This paper is devoted to show how the already widely used scientific computing systems (in our case Maple) integrating symbolic and numeric capabilities can be used to develop problem solving environments very useful to solve problems into a CAD/CAM framework before they are integrated into a concrete CAD/CAM system. It is shown and motivated how a...

In this paper, we describe the application of a new version of Barnett’s method to the squarefree decomposition of a univariate polynomial with coefficients in K [ x ], x being a parameter andK a characteristic zero field. This new version of Barnett’s method uses Bezoutian matrices instead of matrices obtained from evaluating polynomials in a comp...

In this paper we present some personal experiences with Computer Algebra applications to industrial problems. In many cases the involved Com- puter Algebra problems seem as challenging as climbing up a difficult peak. Then one finds out that the trail leads up to a quite rugged hill ... This point of view will be illustrated with "real" examples co...

This paper is devoted to showing how computer algebra and quantifier elimination techniques can be used in order to solve the Birkhoff interpolation problem in the multivariate case in some special situations.

We present a new algorithm for computing the triangular decomposition of a polynomial system of equations with a finite number of complex solutions making it possible to recover, in many cases, the explicit dependence between the coefficients of the initial system and the coefficients of the polynomials in the final one. The main tools used are the...

Distributions with normal conditionals have biquadratic regression functions. Consequently, in contrast to classical bivariate normal distributions, their densities can be multimodal. Criteria for determining the number of modes are discussed and illustrations of representative multimodal densities are provided.

this report is, therefore, to describe our perception of the particular needs of European Industry, which will determine the choice of which algorithms are to be included inside the FRISCO framework.

this report, we explain what are the available technics for dealing with particular cases which are suciently relevant/interesting for being implemented and integrated in the FRISCO framework.