Exact Arrangements on Tori and Dupin Cyclides

An algorithm and implementation is presented to compute the exact arrangement induced by arbitrary algebraic surfaces on a parametrized ring Dupin cyclide. The family of Dupin cyclides contains as a special case the torus. The intersection of an algebraic surface of degree $n$ with a reference cyclide is represented as a real algebraic curve of bi-degree $(2n,2n)$ in the two-dimensional parameter space of the cyclide. We use Eigenwillig and Kerber: ``Exact and Efficient 2D-Arrangements of Arbitrary Algebraic Curves'', SODA~2008, to compute a planar arrangement of such curves and extend their approach to obtain more asymptotic information about curves approaching the boundary of the cyclide's parameter space. With that, we can base our implementation on the general software framework by Berberich~et.~al.: ``Sweeping and Maintaining Two-Dimensional Arrangements on Surfaces: A First Step'', ESA~2007. Our contribution provides the demanded techniques to model the special geometry of surfaces intersecting a cyclide and the special topology of the reference surface of genus one. The contained implementation is complete and does not assume generic position. Our experiments show that the combinatorial overhead of the framework does not harm the efficiency of the method. Our experiments show that the overall performance is strongly coupled to the efficiency of the implementation for arrangements of algebraic plane curves

Minkowski Sum Construction and other Applications of Arrangements of Geodesic Arcs on the Sphere

We present two exact implementations of efficient output-sensitive algorithms that compute Minkowski sums of two convex polyhedra in 3D. We do not assume general position. Namely, we handle degenerate input, and produce exact results. We provide a tight bound on the exact maximum complexity of Minkowski sums of polytopes in 3D in terms of the number of facets of the summand polytopes. The algorithms employ variants of a data structure that represents arrangements embedded on two-dimensional parametric surfaces in 3D, and they make use of many operations applied to arrangements in these representations. We have developed software components that support the arrangement data-structure variants and the operations applied to them. These software components are generic, as they can be instantiated with any number type. However, our algorithms require only (exact) rational arithmetic. These software components together with exact rational-arithmetic enable a robust, efficient, and elegant implementation of the Minkowski-sum constructions and the related applications. These software components are provided through a package of the Computational Geometry Algorithm Library (CGAL) called Arrangement_on_surface_2. We also present exact implementations of other applications that exploit arrangements of arcs of great circles embedded on the sphere. We use them as basic blocks in an exact implementation of an efficient algorithm that partitions an assembly of polyhedra in 3D with two hands using infinite translations. This application distinctly shows the importance of exact computation, as imprecise computation might result with dismissal of valid partitioning-motions.Comment: A Ph.D. thesis carried out at the Tel-Aviv university. 134 pages long. The advisor was Prof. Dan Halperi

Real root isolation for exact and approximate polynomials using descartes' rule of signs

Collins und Akritas (1976) have described the Descartes method for isolating the real roots of an integer polynomial in one variable. This method recursively subdivides an initial interval until Descartes' Rule of Signs indicates that all roots have been isolated. The partial converse of Descartes' Rule by Obreshkoff (1952) in conjunction with the bound of Mahler (1964) and Davenport (1985) leads us to an asymptotically almost tight bound for the resulting subdivision tree. It implies directly the best known complexity bounds for the equivalent forms of the Descartes method in the power basis (Collins/Akritas, 1976), the Bernstein basis (Lane/Riesenfeld, 1981) and the scaled Bernstein basis (Johnson, 1991), which are presented here in a unified fashion. Without losing correctness of the output, we modify the Descartes method such that it can handle bitstream coefficients, which can be approximated arbitrarily well but cannot be determined exactly. We analyze the computing time and precision requirements. The method described elsewhere by the author together with Kerber/Wolpert (2007) and Kerber (2008) to determine the arrangement of plane algebraic curves rests in an essential way on variants of the bitstream Descartes algorithm; we analyze a central part of it.Collins und Akritas (1976) haben das Descartes-Verfahren zur Einschließung der reellen Nullstellen eines ganzzahligen Polynoms in einer Veränderlichen angegeben. Das Verfahren unterteilt rekursiv ein Ausgangsintervall, bis die Descartes'sche Vorzeichenregel anzeigt, dass alle Nullstellen getrennt worden sind. Die partielle Umkehrung der Descartes'schen Regel nach Obreschkoff (1952) in Verbindung mit der Schranke von Mahler (1964) und Davenport (1985) führt uns auf eine asymptotisch fast scharfe Schranke für den sich ergebenden Unterteilungsbaum. Daraus folgen direkt die besten bekannten Komplexitätsschranken für die äquivalenten Formen des Descartes-Verfahrens in der Monom-Basis (Collins/Akritas, 1976), der Bernstein-Basis (Lane/Riesenfeld, 1981) und der skalierten Bernstein-Basis (Johnson, 1991), die hier vereinheitlicht dargestellt werden. Ohne dass die Korrektheit der Ausgabe verloren geht, modifizieren wir das Descartes-Verfahren so, dass es mit "Bitstream"-Koeffizienten umgehen kann, die beliebig genau angenähert, aber nicht exakt bestimmt werden können. Wir analysieren die erforderliche Rechenzeit und Präzision. Das vom Verfasser mit Kerber/Wolpert (2007) und Kerber (2008) an anderer Stelle beschriebene Verfahren zur Bestimmung des Arrangements (der Schnittfigur) ebener algebraischer Kurven fußt wesentlich auf Varianten des Bitstream-Descartes-Verfahrens; wir analysieren einen zentralen Teil davon

Collection of abstracts of the 24th European Workshop on Computational Geometry

International audienceThe 24th European Workshop on Computational Geomety (EuroCG'08) was held at INRIA Nancy - Grand Est & LORIA on March 18-20, 2008. The present collection of abstracts contains the 63 scientific contributions as well as three invited talks presented at the workshop

Geometric algorithms for algebraic curves and surfaces

This work presents novel geometric algorithms dealing with algebraic curves and surfaces of arbitrary degree. These algorithms are exact and complete — they return the mathematically true result for all input instances. Efficiency is achieved by cutting back expensive symbolic computation and favoring combinatorial and adaptive numerical methods instead, without spoiling exactness in the overall result. We present an algorithm for computing planar arrangements induced by real algebraic curves. We show its efficiency both in theory by a complexity analysis, as well as in practice by experimental comparison with related methods. For the latter, our solution has been implemented in the context of the Cgal library. The results show that it constitutes the best current exact implementation available for arrangements as well as for the related problem of computing the topology of one algebraic curve. The algorithm is also applied to related problems, such as arrangements of rotated curves, and arrangments embedded on a parameterized surface. In R3, we propose a new method to compute an isotopic triangulation of an algebraic surface. This triangulation is based on a stratification of the surface, which reveals topological and geometric information. Our implementation is the first for this problem that makes consequent use of numerical methods, and still yields the exact topology of the surface.Diese Arbeit stellt neue Algorithmen für algebraische Kurven und Flächen von beliebigem Grad vor. Diese Algorithmen liefern für alle Eingaben das mathematisch korrekte Ergebnis. Wir erreichen Effizienz, indem wir aufwendige symbolische Berechnungen weitesgehend vermeiden, und stattdessen kombinatorische und adaptive numerische Methoden einsetzen, ohne die Exaktheit des Resultats zu zerstören. Der Hauptbeitrag ist ein Algorithmus zur Berechnung von planaren Arrangements, die durch reelle algebraische Kurven induziert sind. Wir weisen die Effizienz des Verfahrens sowohl theoretisch durch eine Komplexitätsanalyse, als auch praktisch durch experimentelle Vergleiche nach. Dazu haben wir unser Verfahren im Rahmen der Softwarebibliothek Cgal implementiert. Die Resultate belegen, dass wir die zur Zeit beste verfügbare exakte Software bereitstellen. Der Algorithmus wird zur Arrangementberechnung rotierter Kurven, oder für Arrangements auf parametrisierten Oberflächen eingesetzt. Im R3 geben wir ein neues Verfahren zur Berechnung einer isotopen Triangulierung einer algebraischen Oberfläche an. Diese Triangulierung basiert auf einer Stratifizierung der Oberfläche, die topologische und geometrische Informationen berechnet. Unsere Implementierung ist die erste für dieses Problem, welche numerische Methoden konsequent einsetzt, und dennoch die exakte Topologie der Oberfläche liefert