Analysis of the Incircle predicate for the Euclidean Voronoi diagram of axes-aligned line segments

In this paper we study the most-demanding predicate for computing the Euclidean Voronoi diagram of axes-aligned line segments, namely the Incircle predicate. Our contribution is two-fold: firstly, we describe, in algorithmic terms, how to compute the Incircle predicate for axes-aligned line segments, and secondly we compute its algebraic degree. Our primary aim is to minimize the algebraic degree, while, at the same time, taking into account the amount of operations needed to compute our predicate of interest. In our predicate analysis we show that the Incircle predicate can be answered by evaluating the signs of algebraic expressions of degree at most 6; this is half the algebraic degree we get when we evaluate the Incircle predicate using the current state-of-the-art approach. In the most demanding cases of our predicate evaluation, we reduce the problem of answering the Incircle predicate to the problem of computing the sign of the value of a linear polynomial (in one variable), when evaluated at a known specific root of a quadratic polynomial (again in one variable). Another important aspect of our approach is that, from a geometric point of view, we answer the most difficult case of the predicate via implicitly performing point locations on an appropriately defined subdivision of the place induced by the Voronoi circle implicated in the Incircle predicate.Comment: 17 pages, 4 figures, work presented in the paper is part of M. Kamarianakis' M.S. thesi

Classroom Examples of Robustness Problems in Geometric Computations

International audienceThe algorithms of computational geometry are designed for a machine model with exact real arithmetic. Substituting floating point arithmetic for the assumed real arithmetic may cause implementations to fail. Although this is well known, there is no comprehensive documentation of what can go wrong and why. In this extended abstract, we study a simple incremental algorithm for planar convex hulls and give examples which make the algorithm fail in all possible ways. We also show how to construct failure-examples semi-systematically and discuss the geometry of the floating point implementation of the orientation predicate. We hope that our work will be useful for teaching computational geometry. The full paper is available at http://hal.inria.fr/inria-00344310/. It contains further examples, more theory, and color pictures. We strongly recommend to read the full paper instead of this extended abstract

General Analysis Tool Box for Controlled Perturbation

The implementation of reliable and efficient geometric algorithms is a challenging task. The reason is the following conflict: On the one hand, computing with rounded arithmetic may question the reliability of programs while, on the other hand, computing with exact arithmetic may be too expensive and hence inefficient. One solution is the implementation of controlled perturbation algorithms which combine the speed of floating-point arithmetic with a protection mechanism that guarantees reliability, nonetheless. This paper is concerned with the performance analysis of controlled perturbation algorithms in theory. We answer this question with the presentation of a general analysis tool box. This tool box is separated into independent components which are presented individually with their interfaces. This way, the tool box supports alternative approaches for the derivation of the most crucial bounds. We present three approaches for this task. Furthermore, we have thoroughly reworked the concept of controlled perturbation in order to include rational function based predicates into the theory; polynomial based predicates are included anyway. Even more we introduce object-preserving perturbations. Moreover, the tool box is designed such that it reflects the actual behavior of the controlled perturbation algorithm at hand without any simplifying assumptions.Comment: 90 pages, 30 figure

General analysis tool box for controlled perturbation algorithms and complexity and computation of Θ-guarded regions

This thesis belongs to the field of computational geometry and addresses the following two issues. 1. The implementation of reliable and efficient geometric algorithms is a challenging task. Controlled perturbation combines the speed of floating-point arithmetic with a mechanism that guarantees reliability. We present a general tool box for the analysis of controlled perturbation algorithms. This tool box is separated into independent components. We present three alternative approaches for the derivation of the most important bounds. Furthermore, we have included polynomial-based predicates, rational function-based predicates, and object-preserving perturbations into the theory. Moreover, the tool box is designed such that it reflects the actual behavior of the algorithm at hand without simplifying assumptions. 2. Illumination and guarding problems are a wide field in computational and combinatorial geometry to which we contribute the complexity and computation of Θ-guarded regions. They are a generalization of the convex hull and are related to α-hulls and Θ-maxima. The difficulty in the study of Θ-guarded regions is the dependency of its shape and complexity on Θ. For all angles Θ, we prove fundamental properties of the region, derive lower and upper bounds on its worst-case complexity, and present an algorithm to compute the region.Diese Dissertation auf dem Gebiet der Algorithmischen Geometrie beschäftigt sich mit den folgenden zwei Problemen. 1. Die Implementierung von verlässlichen und effizienten geometrischen Algorithmen ist eine herausfordernde Aufgabe. Controlled Perturbation verknüpft die Geschwindigkeit von Fließkomma-Arithmetik mit einem Mechanismus, der die Verlässlichkeit garantiert. Wir präsentieren einen allgemeinen ,,Werkzeugkasten” zum Analysieren von Controlled Perturbation Algorithmen. Dieser Werkzeugkasten ist in unabhängige Komponenten aufgeteilt. Wir präsentieren drei alternative Methoden für die Herleitung der wichtigsten Schranken. Des Weiteren haben wir alle Prädikate, die auf Polynomen und rationalen Funktionen beruhen, sowie Objekt-erhaltende Perturbationen in die Theorie miteinbezogen. Darüber hinaus wurde der Werkzeugkasten so entworfen, dass er das tatsächliche Verhalten des untersuchten Algorithmus ohne vereinfachende Annahmen widerspiegelt. 2. Illumination und Guarding Probleme stellen ein breites Gebiet der Algorithmischen und Kombinatorischen Geometrie dar. Hierzu tragen wir die Komplexität und Berechnung von Θ-bewachten Regionen bei. Sie stellen eine Verallgemeinerung der konvexen Hülle dar und sind mit α-hulls und Θ-maxima verwandt. Die Schwierigkeit beim Studium der Θ-bewachten Regionen ist die Abhängigkeit ihrer Form und Komplexität von Θ. Für alle Winkel Θ beweisen wir grundlegende Eigenschaften der Region, leiten untere und obere Schranken ihrer worst-case Komplexität her und präsentieren einen Algorithmus, um die Region zu berechnen

General analysis tool box for controlled perturbation algorithms and complexity and computation of Θ-guarded regions

This thesis belongs to the field of computational geometry and addresses the following two issues. 1. The implementation of reliable and efficient geometric algorithms is a challenging task. Controlled perturbation combines the speed of floating-point arithmetic with a mechanism that guarantees reliability. We present a general tool box for the analysis of controlled perturbation algorithms. This tool box is separated into independent components. We present three alternative approaches for the derivation of the most important bounds. Furthermore, we have included polynomial-based predicates, rational function-based predicates, and object-preserving perturbations into the theory. Moreover, the tool box is designed such that it reflects the actual behavior of the algorithm at hand without simplifying assumptions. 2. Illumination and guarding problems are a wide field in computational and combinatorial geometry to which we contribute the complexity and computation of Θ-guarded regions. They are a generalization of the convex hull and are related to α-hulls and Θ-maxima. The difficulty in the study of Θ-guarded regions is the dependency of its shape and complexity on Θ. For all angles Θ, we prove fundamental properties of the region, derive lower and upper bounds on its worst-case complexity, and present an algorithm to compute the region.Diese Dissertation auf dem Gebiet der Algorithmischen Geometrie beschäftigt sich mit den folgenden zwei Problemen. 1. Die Implementierung von verlässlichen und effizienten geometrischen Algorithmen ist eine herausfordernde Aufgabe. Controlled Perturbation verknüpft die Geschwindigkeit von Fließkomma-Arithmetik mit einem Mechanismus, der die Verlässlichkeit garantiert. Wir präsentieren einen allgemeinen ,,Werkzeugkasten” zum Analysieren von Controlled Perturbation Algorithmen. Dieser Werkzeugkasten ist in unabhängige Komponenten aufgeteilt. Wir präsentieren drei alternative Methoden für die Herleitung der wichtigsten Schranken. Des Weiteren haben wir alle Prädikate, die auf Polynomen und rationalen Funktionen beruhen, sowie Objekt-erhaltende Perturbationen in die Theorie miteinbezogen. Darüber hinaus wurde der Werkzeugkasten so entworfen, dass er das tatsächliche Verhalten des untersuchten Algorithmus ohne vereinfachende Annahmen widerspiegelt. 2. Illumination und Guarding Probleme stellen ein breites Gebiet der Algorithmischen und Kombinatorischen Geometrie dar. Hierzu tragen wir die Komplexität und Berechnung von Θ-bewachten Regionen bei. Sie stellen eine Verallgemeinerung der konvexen Hülle dar und sind mit α-hulls und Θ-maxima verwandt. Die Schwierigkeit beim Studium der Θ-bewachten Regionen ist die Abhängigkeit ihrer Form und Komplexität von Θ. Für alle Winkel Θ beweisen wir grundlegende Eigenschaften der Region, leiten untere und obere Schranken ihrer worst-case Komplexität her und präsentieren einen Algorithmus, um die Region zu berechnen

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