Advances in Algebraic Geometric Computation

Lecture Notes in Computer Science, 1997

A plane algebraic curve is given as the zeros of a bivariate polynomial. However, this implicit representation is badly suited for many applications, for instance in computer aided geometric design. What we want in many of these applications is a rational parametrization of an algebraic curve. There are several approaches to deciding whether an algebraic curve is parametrizable and if so computing a parametrization. In all these approaches we ultimately need some simple points on the curve. The eld in which we can nd such points crucially in uences the coe cients in the resulting parametrization. We show how to nd such simple points over some practically interesting elds. Consequently, we are able to decide whether an algebraic curve de ned over the rational numbers can be parametrized over the rationals or the reals. Some of these ideas also apply to parametrization of surfaces. If in the term geometric reasoning we do not only include the process of proving or disproving geometric statements, but also the analysis and manipulation of geometric objects, then algorithms for parametrization play an important role in this wider view of geometric reasoning.

Handbook of Computer Aided Geometric Design, 2002

The concepts and methods of algebra and algebraic geometry have found significant applications in many disciplines. This chapter presents a collection of gleanings from algebra or algebraic geometry that hold practical value for the field of computer aided geometric design. We focus on the insights, algorithm enhancements and practical capabilities that algebraic methods have contributed to CAGD. Specifically, we examine resultants and Gröbner basis, and discuss their applications in implicitization, inversion, parametrization and intersection algorithms. Other topics of CAGD research work using algebraic methods are also outlined.

ACM Communications in Computer Algebra, 2012

I would like to express my gratitude to Maria and Thomas Langer for showing me part of the Austrian way and culture and for making my staying in Linz a pleasant home-like experience. Last but not least I want to especially thank my boyfriend Andreas Langer for his constant understanding and support, for his optimism, for always encouraging me and for his useful suggestions and hints on this thesis.

Journal of Symbolic Computation, 2001

This paper presents an O(n 2) algorithm, based on Gröbner basis techniques, to compute the µ-basis of a degree n planar rational curve. The prior method involved solving a set of linear equations whose complexity by standard numerical methods was O(n 3). The µ-basis is useful in computing the implicit equation of a parametric curve and can express the implicit equation in the form of a determinant that is smaller than that obtained by taking the resultant of the parametric equations.

Computer Aided Geometric Design, 2008

We describe a new subdivision method to efficiently compute the topology and the arrangement of implicit planar curves. We emphasize that the output topology and arrangement are guaranteed to be correct. Although we focus on the implicit case, the algorithm can also treat parametric or piecewise linear curves without much additional work and no theoretical difficulties.

International Journal of Pure and Apllied Mathematics, 2015

Given a parametric representation of an algebraic projective surface S of the ordinary space we give a new algorithm for finding the implicit cartesian equation of S. The algorithm is based on finding a suitable finite number of points on S and computing, by linear algebra, the equation of the surface of least degree that passes through the points.

Proceedings of the international symposium on Symbolic and algebraic computation - ISSAC '94, 1994

In this paper I want to present a new method for computing parametrizations of algebraic curves. Basically this method is a direct application of integral basis computation. Examples show that this method is faster than older methods.

arXiv (Cornell University), 2019

Quadratic surfaces gain more and more attention among the Geometric Algebra community and some frameworks were proposed in order to represent, transform, and intersect these quadratic surfaces. As far as the authors know, none of these frameworks support all the operations required to completely handle these surfaces. Some frameworks do not allow the construction of quadratic surfaces from control points when others do not allow to transform these quadratic surfaces. However, if we consider all the frameworks together, then all the required operations over quadratic are covered. This paper presents a unification of these frameworks that enables to represent any quadratic surfaces either using control points or from the coefficients of its implicit form. The proposed approach also allows to transform any quadratic surfaces and to compute their intersection and to easily extract some geometric properties.

The IMA Volumes in Mathematics and its Applications, 2009

Recently, the visualization of implicitly given algebraic curves and surfaces has become an area of active research. Most of the approaches either use raytracing, subdivision or sweeping techniques to produce a good approximate picture of the varieties, sometimes by using hardware equipment such as graphics processing units. We provide a list of equations of plane curves which may serve as a list of benchmarks for visualization software. In most cases, we give whole series of examples which yield equations for infinitely many degrees. Even for low degrees, there is currently no software which visualizes all examples correctly in real-time, so we call them challenges. For most of the equations in our list, we are able to prove that they are at least close to the most difficult possible ones. For convenience, our list is also available in the form of a text file. Moreoever, the paper includes a brief introduction to some of the terminology from singularity theory for researchers from the computer graphics community because singularities appear frequently when treating complicated cases.

Proceedings of the 2011 International Workshop on Symbolic-Numeric Computation - SNC '11, 2011

We present a new certified and complete algorithm to compute arrangements of real planar algebraic curves. Our algorithm provides a geometric-topological analysis of the decomposition of the plane induced by a finite number of algebraic curves in terms of a cylindrical algebraic decomposition of the plane. Compared to previous approaches, we improve in two main aspects: Firstly, we significantly reduce the amount of exact operations, that is, our algorithms only uses resultant and gcd as purely symbolic operations. Secondly, we introduce a new hybrid method in the lifting step of our algorithm which combines the usage of a certified numerical complex root solver and information derived from the resultant computation. Additionally, we never consider any coordinate transformation and the output is also given with respect to the initial coordinate system. We implemented our algorithm as a prototypical package of the C++-library Cgal. Our implementation exploits graphics hardware to expedite the resultant and gcd computation. We also compared our implementation with the current reference implementation, that is, Cgal's curve analysis and arrangement for algebraic curves. For various series of challenging instances, our experiments show that the new implementation outperforms the existing one.

Proceedings Visualization '94, 1994

We present a comprehensive algorithm to construct a topologically correct triangulation of the real affine part of a rational parametric surface with few restrictions on the defining rational functions. The rational functions are allowed to be undefined on domain curves (pole curves) and at certain special points (base points), and the surface is allowed to have nodal or cuspidal self-intersections.

Computer Aided Geometric Design, 2013

We introduce the notion of radical parametrization of a surface, and we provide algorithms to compute such type of parametrizations for families of surfaces, like: Fermat surfaces, surfaces with a high multiplicity (at least the degree minus 4) singularity, all irreducible surfaces of degree at most 5, all irreducible singular surfaces of degree 6, and surfaces containing a pencil of low-genus curves. In addition, we prove that radical parametrizations are preserved under certain type of geometric constructions that include offset and conchoids.

Algorithms and computation in mathematics, 2008

In this paper we give an overview of the symbolic computation of rational parametrizations for algebraic curves. Parametrization of algebraic curves has been a topic in algebraic geometry for a long time and there are basically two approaches to the problem. The first approach is a geometric one, whereby a parametrization is determined from the intersection points of the given curve with a certain linear system of algebraic curves. The second one is an algebraic one involving the computation of integral bases. Our own research has mainly been on the first of these approaches, so will stress the geometric approach.

We introduce a new formalism and a number of new results in the context of geometric computational vision. The classical scope of the research in geometric computer vision is essentially limited to static configurations of points and lines in $P^3$ . By using some well known material from algebraic geometry, we open new branches to computational vision. We introduce algebraic curves embedded in $P^3$ as the building blocks from which the tensor of a couple of cameras (projections) can be computed. In the process we address dimensional issues and as a result establish the minimal number of algebraic curves required for the tensor variety to be discrete as a function of their degree and genus. We then establish new results on the reconstruction of an algebraic curves in $P^3$ from multiple projections on projective planes embedded in $P^3$ . We address three different presentations of the curve: (i) definition by a set of equations, for which we show that for a generic configuration, ...