An Elimination Method for Solving Bivariate Polynomial Systems: Eliminating the Usual Drawbacks

We present an exact and complete algorithm to isolate the real solutions of a zero-dimensional bivariate polynomial system. The proposed algorithm constitutes an elimination method which improves upon existing approaches in a number of points. First, the amount of purely symbolic operations is significantly reduced, that is, only resultant computation and square-free factorization is still needed. Second, our algorithm neither assumes generic position of the input system nor demands for any change of the coordinate system. The latter is due to a novel inclusion predicate to certify that a certain region is isolating for a solution. Our implementation exploits graphics hardware to expedite the resultant computation. Furthermore, we integrate a number of filtering techniques to improve the overall performance. Efficiency of the proposed method is proven by a comparison of our implementation with two state-of-the-art implementations, that is, LPG and Maple's isolate. For a series of challenging benchmark instances, experiments show that our implementation outperforms both contestants.Comment: 16 pages with appendix, 1 figure, submitted to ALENEX 201

Deconstructing Approximate Offsets

We consider the offset-deconstruction problem: Given a polygonal shape Q with n vertices, can it be expressed, up to a tolerance \eps in Hausdorff distance, as the Minkowski sum of another polygonal shape P with a disk of fixed radius? If it does, we also seek a preferably simple-looking solution P; then, P's offset constitutes an accurate, vertex-reduced, and smoothened approximation of Q. We give an O(n log n)-time exact decision algorithm that handles any polygonal shape, assuming the real-RAM model of computation. A variant of the algorithm, which we have implemented using CGAL, is based on rational arithmetic and answers the same deconstruction problem up to an uncertainty parameter \delta; its running time additionally depends on \delta. If the input shape is found to be approximable, this algorithm also computes an approximate solution for the problem. It also allows us to solve parameter-optimization problems induced by the offset-deconstruction problem. For convex shapes, the complexity of the exact decision algorithm drops to O(n), which is also the time required to compute a solution P with at most one more vertex than a vertex-minimal one.Comment: 18 pages, 11 figures, previous version accepted at SoCG 2011, submitted to DC

Robust and efficient software for problems in 2.5-dimensional non-linear geometry : algorithms and implementations

We discuss how to compute and implement three geometric problems dealing with nonlinear three-dimensional surfaces. As a main tool we rely on planar subdivisions induced by algebraic curves, developed in CGAL (Computational Geometry Algorithm Library). First, we achieve lower envelopes of quadrics using CGAL'S Envelope_3 package. Second, we extend CGAL's Arrangement_2 package to support two-dimensional arrangements on a parametric reference surface. Two main examples are discussed: Arrangements induced by algebraic surfaces on an elliptic quadric and on a ring Dupin cyclide. Third, we decompose a set of quadrics or a set of algebraic surfaces into cells using projection. Our goal is to achieve topological information for the surfaces, while preserving their geometric properties. We maintain a special two-dimensional arrangement; the lifting to the third dimension benefits from the recently presented bitstream Descartes method. The obtained cell decomposition supports a set of other geometric applications on surfaces. Our implementations follow the geometric programming paradigm. That is, we split combinatorial tasks from geometric operations by generic programming techniques. It is also ensured that each geometric predicate returns the mathematically correct result, even if it internally exploits approximative methods to speed up the computation. The thesis is written in English.Wir besprechen die Berechnung und Implementierung dreier Probleme aus der algorithmischen Geometrie, deren Eingabe aus gekrümmten Oberflächen besteht. Als Werkzeug benutzen wir in CGAL (Computational Geometry Algorithm Library) entwickelte Zerlegungen der Ebene durch algebraische Kurven. Zunächst berechnen wir die untere Einhüllende einer Menge von Quadriken. Danach erweitern wir CGALs Arrangement_2 Paket, so dass zweidimensionale Zerlegungen auf para-meterisierbaren Oberflächen berechnet werden können, und führen zwei konkrete Beispiele aus: Zerlegungen induziert durch algebraische Oberflächen auf einer Quadrik und auf einem ringförmigen Zykliden nach Dupin. Zum Abschluss unterteilen wir eine Menge von Quadriken bzw. algebraischen Oberflächen in disjunkte Untermannigfaltigkeiten mit Hilfe einer Projektion. Die Hebung erfolgt mit einem kürzlich vorgestellten approximativen Verfahren zur Nullstellenisolation (bitstream Descartes). Ingesamt erhalten wir geometrische Eigenschaften der Eingabe und erfahren mehr über deren topologische Zusammensetzung. Die kombinatorische Ausgabe hilft bei der Berechnung anderer geometrischer Probleme auf den Oberflächen. Unsere Implementierungen trennen kombinatorische Aufgaben von geometrischen durch Anwenden von generischen Programmiertechniken. Wir stellen außerdem sicher, dass Prädikate stets das mathematisch korrekte Ergebnis ausgeben, auch wenn sie intern mit approximativen Methoden rechnen. Die Arbeit ist in englischer Sprache verfasst

Exact Symbolic-Numeric Computation of Planar Algebraic Curves

We present a novel certified and complete algorithm to compute arrangements of real planar algebraic curves. It 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. From a high-level perspective, the overall method splits into two main subroutines, namely an algorithm denoted Bisolve to isolate the real solutions of a zero-dimensional bivariate system, and an algorithm denoted GeoTop to analyze a single algebraic curve. Compared to existing approaches based on elimination techniques, we considerably improve the corresponding lifting steps in both subroutines. As a result, generic position of the input system is never assumed, and thus our algorithm never demands for any change of coordinates. In addition, we significantly limit the types of involved exact operations, that is, we only use resultant and gcd computations as purely symbolic operations. The latter results are achieved by combining techniques from different fields such as (modular) symbolic computation, numerical analysis and algebraic geometry. We have implemented our algorithms as prototypical contributions to the C++-project CGAL. They exploit graphics hardware to expedite the symbolic computations. We have also compared our implementation with the current reference implementations, that is, LGP and Maple's Isolate for polynomial system solving, and CGAL's bivariate algebraic kernel for analyses and arrangement computations of algebraic curves. For various series of challenging instances, our exhaustive experiments show that the new implementations outperform the existing ones.Comment: 46 pages, 4 figures, submitted to Special Issue of TCS on SNC 2011. arXiv admin note: substantial text overlap with arXiv:1010.1386 and arXiv:1103.469

Exact geometric-topological analysis of algebraic surfaces

We present a method to compute the exact topology of a real algebraic surface $S$, implicitly given by a polynomial $f \in \mathbb{Q}[x,y,z]$ of arbitrary degree $N$. Additionally, our analysis provides geometric information as it supports the computation of arbitrary precise samples of $S$ including critical points. We use a projection approach, similar to Collins' cylindrical algebraic decomposition (cad). In comparison we reduce the number of output cells to $O(N^5)$ by constructing a special planar arrangement instead of a full cad in the projection plane. Furthermore, our approach applies numerical and combinatorial methods to minimize costly symbolic computations. The algorithm handles all sorts of degeneracies without transforming the surface into a generic position. We provide a complete implementation of the algorithm, written in C++. It shows good performance for many well known examples from algebraic geometry