arXiv subjects

Juan Gerardo Alcázar

Publications and source records attributed to Juan Gerardo Alcázar.

18 recordsLinked to original sources

arXiv2024

Nonzero Constant Wronskians of Polynomials and Laurent Polynomials, and Geometric Consequences

We characterize the polynomials $p_1(t), ... , p_n(t)$ whose Wronskian $W(p_1, ... , p_n)$ is a nonzero constant. Then, we generalize our results to characterize the Laurent polynomials with the same property. Finally, for rational functions we prove an impossibility result for $n=2$, and pose the case $n \geq 3$ as an open question, although we suggest an impossibility conjecture. Some geometric consequences are derived, especially in the case of polynomials.

math.AG↗

arXiv2024

Detecting affine equivalences between certain types of parametric curves, in any dimension

Two curves are affinely equivalent if there exists an affine mapping transforming one of them onto the other. Thus, detecting affine equivalence comprises, as important particular cases, similarity, congruence and symmetry detection. In this paper we generalize previous results by the authors to provide an algorithm for computing the affine equivalences between two parametric curves of certain types, in any dimension. In more detail, the algorithm is valid for rational curves, and for parametric curves with non-rational but meromorphic components admitting a rational inverse. Unlike other algorithms already known for rational curves, the algorithm completely avoids polynomial system solving, and uses bivariate factoring, instead, as a fundamental tool. The algorithm has been implemented in the computer algebra system {\tt Maple}, and can be freely downloaded and used.

math.AG↗

arXiv2022

A new method to detect projective equivalences and symmetries of rational $3D$ curves

We present a new approach using differential invariants to detect projective equivalences and symmetries between two rational parametric $3D$ curves properly parametrized. In order to do this, we introduce two differential invariants that commute with Möbius transformations, which are the transformations in the parameter space associated with the projective equivalences between the curves. The Möbius transformations are found by first computing the gcd of two polynomials built from the differential invariants, and then searching for the Möbius-like factors of this gcd. The projective equivalences themselves are easily computed from the Möbius transformations. In particular, and unlike previous approaches, we avoid solving big polynomial systems. The algorithm has been implemented in Maple, and evidences of its efficiency as well as a comparison with previous approaches are given.

math.AG↗

arXiv2021

Affine equivalences of surfaces of translation and minimal surfaces, and applications to symmetry detection and design

We introduce a characterization for affine equivalence of two surfaces of translation defined by either rational or meromorphic generators. In turn, this induces a similar characterization for minimal surfaces. In the rational case, our results provide algorithms for detecting affine equivalence of these surfaces, and therefore, in particular, the symmetries of a surface of translation or a minimal surface of the considered types. Additionally, we apply our results to designing surfaces of translation and minimal surfaces with symmetries, and to computing the symmetries of the higher-order Enneper surfaces.

math.AG↗

arXiv2019

Computing the topology of a planar or space hyperelliptic curve

We present algorithms to compute the topology of 2D and 3D hyperelliptic curves. The algorithms are based on the fact that 2D and 3D hyperelliptic curves can be seen as the image of a planar curve (the Weierstrass form of the curve), whose topology is easy to compute, under a birational mapping of the plane or the space. We report on a {\tt Maple} implementation of these algorithms, and present several examples. Complexity and certification issues are also discussed.

math.GT↗

arXiv2019

Affine differential geometry and affine rotation surfaces: algebraic surfaces invariant under non-Euclidean affine rotations

Affine rotation surfaces are a generalization of the well-known surfaces of revolution. Affine rotation surfaces arise naturally within the framework of affine differential geometry, a field started by Blaschke in the first decades of the past century. Affine rotations are the affine equivalents of Euclidean rotations, and include certain shears as well as Euclidean rotations. Affine rotation surfaces are surfaces invariant under affine rotations. In this paper, we analyze several properties of algebraic affine rotation surfaces and, by using some notions and results from affine differential geometry, we develop an algorithm for determining whether or not an algebraic surface given in implicit form, or in some cases in rational parametric form, is an affine rotation surface. We also show how to find the axis of an affine rotation surface. Additionally, we discuss several properties of affine spheres, analogues of Euclidean spheres in the context of affine differential geometry.

math.AG↗

arXiv2019

Estimation of spatio-temporal wave grouping properties using Delaunay triangulation and spline techniques

Wave groups can be detected and studied by using the wave envelope. So far, the method used to compute the wave envelope employs the Riesz transform. However, such a technique always produces symmetric envelopes, which is only realistic in the case of linear waves. In this paper we present a new method to compute the wave envelope providing more realistic results. In particular, the method allows to detect non-symmetry in the wave envelope, something useful, for instance, when detecting groups of high waves. The method computes first the local maxima and minima of the sea surface, and then determines the wave envelope by combining discrete methods, namely the use of the Delaunay triangulation, and tensor-product splines. The proposed method has been applied to simulated wave fields, and also to wave elevations data measured by an X-band radar. The obtained results correctly reproduce the behavior of the simulated waves.

physics.ao-ph↗

arXiv2019

Affine equivalences, isometries and symmetries of ruled rational surfaces

A method is presented for computing all the affine equivalences between two rational ruled surfaces defined by rational parametrizations that works directly in parametric rational form, i.e. without computing or making use of the implicit equation of the surface. The method translates the problem into parameter space and relies on polynomial system solving. Geometrically, the problem is related to finding the projective equivalences between two projective curves (corresponding to the directions of the rulings of the surfaces). This problem was recently addressed in a paper by Hauer and Jüttler, and we exploit the ideas by these authors in the algorithm presented in this paper. The general idea is adapted to computing the isometries between two rational ruled surfaces, and the symmetries of a given rational ruled surface. The efficiency of the method is shown through several examples.

math.AC↗

arXiv2018

On the computation of the straight lines contained in a rational surface

In this paper we present an algorithm to compute the (real and complex) straight lines contained in a rational surface, defined by a rational parameterization. The algorithm relies on the well-known theorem of Differential Geometry that characterizes real straight lines contained in a surface as curves that are simultaneously asymptotic lines, and geodesics. We also report on an implementation carried out in Maple 18, and we compare the behavior of our algorithm with two brute-force approaches.

math.AG↗

arXiv2018

Symmetries and similarities of planar algebraic curves using harmonic polynomials

We present novel, deterministic, efficient algorithms to compute the symmetries of a planar algebraic curve, implicitly defined, and to check whether or not two given implicit planar algebraic curves are similar, i.e. equal up to a similarity transformation. Both algorithms are based on the fact, well-known in Harmonic Analysis, that the Laplacian operator commutes with orthogonal transformations, and on efficient algorithms to find the symmetriessimilarities of a harmonic algebraic curvetwo given harmonic algebraic curves. In fact, we show that in general the problem can be reduced to the harmonic case, except for some special cases, easy to treat.

math.AG↗

arXiv2017

Symmetries of Canal Surfaces and Dupin Cyclides

We develop a characterization for the existence of symmetries of canal surfaces defined by a rational spine curve and rational radius function. In turn, this characterization inspires an algorithm for computing the symmetries of such canal surfaces. For Dupin cyclides in canonical form, we apply the characterization to derive an intrinsic description of their symmetries and symmetry groups, which gives rise to a method for computing the symmetries of a Dupin cyclide not necessarily in canonical form. As a final application, we discuss the construction of patches and blends of rational canal surfaces with a prescribed symmetry.

math.AG↗

arXiv2017

Similarity detection of rational space curves

We provide an algorithm to check whether two rational space curves are related by a similarity. The algorithm exploits the relationship between the curvatures and torsions of two similar curves, which is formulated in a computer algebra setting. Helical curves, where curvature and torsion are proportional, need to be distinguished as a special case. The algorithm is easy to implement, as it involves only standard computer algebra techniques, such as greatest common divisors and resultants, and Gröbner basis for the special case of helical curves. Details on the implementation and experimentation carried out using the computer algebra system Maple 18 are provided.

math.AG↗

arXiv2016

Recognizing projections of rational curves

Given two rational, properly parametrized space curves ${\mathcal C}_1$ and ${\mathcal C}_2$, where $\CCC_2$ is contained in some plane $Π$, we provide an algorithm to check whether or not there exist perspective or parallel projections mapping $\CCC_1$ onto $\CCC_2$, i.e. to recognize $\CCC_2$ as the projection of $\CCC_1$. In the affirmative case, the algorithm provides the eye point(s) of the perspective transformation(s), or the direction(s) of the parallel projection(s). The problem is mainly discussed from a symbolic point of view, but an approximate algorithm is also included.

math.AG↗

arXiv2015

On the square-freeness of the offset equation to a rational planar curve

It is well known that an implicit equation of the offset to a rational planar curve can be computed by removing the extraneous components of the resultant of two certain polynomials computed from the parametrization of the curve. Furthermore, it is also well known that the implicit equation provided by the non-extraneous component of this resultant has at most two irreducible factors. In this paper, we complete the algebraic description of this resultant by showing that the multiplicity of the factors corresponding to the offset can be computed in advance. In particular, when the parametrization is proper, i.e. when the curve is just traced once by the parametrization, we prove that any factor corresponding to a simple component of the offset has multiplicity 1, while the factor corresponding to the special component, if any, has multiplicity 2. Hence, if the parametrization is proper and there is no special component, the non-extraneous part of the resultant is square-free. In fact, this condition is proven to be also sufficient. Additionally, this result provides a simple test to check whether or not a given rational curve is the offset of another curve.

math.AG↗

arXiv2015

A new method to compute the singularities of offsets to rational plane curves

Given a planar curve defined by means of a real rational parametrization, we prove that the affine values of the parameter generating the real singularities of the offset are real roots of a univariate polynomial that can be derived from the parametrization of the original curve, without computing or making use of the implicit equation of the offset. By using this result, a finite set containing all the real singularities of the offset, and in particular all the real self-intersections of the offset, can be computed. We also report on experiments carried out in the computer algebra system Maple, showing the efficiency of the algorithm for moderate degrees.

math.AG↗

arXiv2015

On the Problem of Detecting When Two Implicit Plane Algebraic Curves Are Similar

We make use of the complex implicit representation in order to provide a deterministic algorithm for checking whether or not two implicit algebraic curves are related by a similarity, a central question in Pattern Recognition and Computer Vision. The algorithm has been implemented in the computer algebra system Maple 2015. Examples and evidence of the good practical performance of the algorithm are given.

math.AG↗

arXiv2015

Symmetry Detection of Rational Space Curves from their Curvature and Torsion

We present a novel, deterministic, and efficient method to detect whether a given rational space curve is symmetric. By using well-known differential invariants of space curves, namely the curvature and torsion, the method is significantly faster, simpler, and more general than an earlier method addressing a similar problem. To support this claim, we present an analysis of the arithmetic complexity of the algorithm and timings from an implementation in Sage.

math.AG↗

arXiv2013

Detecting Similarity of Rational Plane Curves

A novel and deterministic algorithm is presented to detect whether two given rational plane curves are related by means of a similarity, which is a central question in Pattern Recognition. As a by-product it finds all such similarities, and the particular case of equal curves yields all symmetries. A complete theoretical description of the method is provided, and the method has been implemented and tested in the Sage system for curves of moderate degrees.

math.AG↗