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

Topologically reliable approximation of curves and surfaces

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Ocean Engineering, 1997.Includes bibliographical references (p. [213]-222).by Wonjoon Cho.Ph.D

Automatic motion of manipulator using sampling based motion planning algorithms - application in service robotics

The thesis presents new approaches for autonomous motion execution of a robotic arm. The calculation of the motion is called motion planning and requires the computation of robot arm's path. The text covers the calculation of the path and several algorithms have been therefore implemented and tested in several real scenarios. The work focuses on sampling based planners, which means that the path is created by connecting explicitly random generated points in the free space. The algorithms can be divided into three categories: those that are working in configuration space(C-Space)(C- Space is the set of all possible joint angles of a robotic arm) , the mixed approaches using both Cartesian and C-Space and those that are using only the Cartesian space. Although Cartesian space seems more appropriate, due to dimensionality, this work illustrates that the C-Space planners can achieve comparable or better results. Initially an enhanced approach for efficient collision detection in C-Space, used by the planners, is presented. Afterwards the N dimensional cuboid region, notated as Rq, is defined. The Rq configures the C-Space so that the sampling is done close to a selected, called center, cell. The approach is enhanced by the decomposition of the Cartesian space into cells. A cell is selected appropriately if: (a) is closer to the target position and (b) lies inside the constraints. Inverse kinematics(IK) are applied to calculate a centre configuration used later by the Rq. The CellBiRRT is proposed and combines all the features. Continuously mixed approaches that do not require goal configuration or an analytic solution of IK are presented. Rq regions as well as Cells are also integrated in these approaches. A Cartesian sampling based planner using quaternions for linear interpolation is also proposed and tested. The common feature of the so far algorithms is the feasibility which is normally against the optimality. Therefore an additional part of this work deals with the optimality of the path. An enhanced approach of CellBiRRT, called CellBiRRT*, is developed and promises to compute shorter paths in a reasonable time. An on-line method using both CellBiRRT and CellBiRRT* is proposed where the path of the robot arm is improved and recalculated even if sudden changes in the environment are detected. Benchmarking with the state of the art algorithms show the good performance of the proposed approaches. The good performance makes the algorithms suitable for real time applications. In this work several applications are described: Manipulative skills, an approach for an semi-autonomous control of the robot arm and a motion planning library. The motion planning library provides the necessary interface for easy use and further development of the motion planning algorithms. It can be used as the part connecting the manipulative skill designing and the motion of a robotic arm

Hierarchical Graphs as Organisational Principle and Spatial Model Applied to Pedestrian Indoor Navigation

In this thesis, hierarchical graphs are investigated from two different angles – as a general modelling principle for (geo)spatial networks and as a practical means to enhance navigation in buildings. The topics addressed are of interest from a multi-disciplinary point of view, ranging from Computer Science in general over Artificial Intelligence and Computational Geometry in particular to other fields such as Geographic Information Science. Some hierarchical graph models have been previously proposed by the research community, e.g. to cope with the massive size of road networks, or as a conceptual model for human wayfinding. However, there has not yet been a comprehensive, systematic approach for modelling spatial networks with hierarchical graphs. One particular problem is the gap between conceptual models and models which can be readily used in practice. Geospatial data is commonly modelled - if at all - only as a flat graph. Therefore, from a practical point of view, it is important to address the automatic construction of a graph hierarchy based on the predominant data models. The work presented deals with this problem: an automated method for construction is introduced and explained. A particular contribution of my thesis is the proposition to use hierarchical graphs as the basis for an extensible, flexible architecture for modelling various (geo)spatial networks. The proposed approach complements classical graph models very well in the sense that their expressiveness is extended: various graphs originating from different sources can be integrated into a comprehensive, multi-level model. This more sophisticated kind of architecture allows for extending navigation services beyond the borders of one single spatial network to a collection of heterogeneous networks, thus establishing a meta-navigation service. Another point of discussion is the impact of the hierarchy and distribution on graph algorithms. They have to be adapted to properly operate on multi-level hierarchies. By investigating indoor navigation problems in particular, the guiding principles are demonstrated for modelling networks at multiple levels of detail. Complex environments like large public buildings are ideally suited to demonstrate the versatile use of hierarchical graphs and thus to highlight the benefits of the hierarchical approach. Starting from a collection of floor plans, I have developed a systematic method for constructing a multi-level graph hierarchy. The nature of indoor environments, especially their inherent diversity, poses an additional challenge: among others, one must deal with complex, irregular, and/or three-dimensional features. The proposed method is also motivated by practical considerations, such as not only finding shortest/fastest paths across rooms and floors, but also by providing descriptions for these paths which are easily understood by people. Beyond this, two novel aspects of using a hierarchy are discussed: one as an informed heuristic exploiting the specific characteristics of indoor environments in order to enhance classical, general-purpose graph search techniques. At the same time, as a convenient by- product of this method, clusters such as sections and wings can be detected. The other reason is to better deal with irregular, complex-shaped regions in a way that instructions can also be provided for these spaces. Previous approaches have not considered this problem. In summary, the main results of this work are: • hierarchical graphs are introduced as a general spatial data infrastructure. In particular, this architecture allows us to integrate different spatial networks originating from different sources. A small but useful set of operations is proposed for integrating these networks. In order to work in a hierarchical model, classical graph algorithms are generalised. This finding also has implications on the possible integration of separate navigation services and systems; • a novel set of core data structures and algorithms have been devised for modelling indoor environments. They cater to the unique characteristics of these environments and can be specifically used to provide enhanced navigation in buildings. Tested on models of several real buildings from our university, some preliminary but promising results were gained from a prototypical implementation and its application on the models

Pfadverfolgungsprobleme aus der Dynamischen Geometrie

Dynamic Geometry is the field of interactively doing geometric constructions using a computer. Usually, the classical ruler-and-compass constructions are considered. The available tools are simulated by the computer. A Dynamic Geometry System is a system to do geometric constructions that has a drag mode. In the drag mode, geometric elements with at least one degree of freedom can be moved, and the remaining part of the geometric construction adjusts automatically. Thus, the computer has to trace the paths of the involved geometric objects during the motion. In this thesis, we focus on the beautiful model by Kortenkamp and Richter-Gebert that is the foundation of the geometry software Cinderella. We embed an algebraic variant of this model into different fields of pure and applied mathematics, which leads to different approaches for realizing the drag mode practically. We develop a numerical method to solve the Tracing Problem that is based on a generic Predictor- Corrector method. Like most numerical methods, this method cannot guarantee the correctness of the computed solution curve, hence ambiguities are not treated satisfactorily. To overcome this problem, we develope a second algorithm that uses interval analysis. This algorithm is robust, and the computed step length is small enough to break up all ambiguities. Critical points are bypassed by detours, where the geometric objects or the corresponding variables in the algebraic model can have complex coordinates. Here, the final configuration depends essentially on the chosen detour, but this procedure due to Kortenkamp and Richter-Gebert leads to a consistent treatment of degeneracies. We investigate the connection of the used model for Dynamic Geometry to Riemann surfaces of algebraic functions.Unter dynamischer Geometrie versteht man das interaktive Erstellen von geometrischen Konstruktionen am Computer. Ein Dynamisches Geometrie System ist ein Geometriesystem, in dem es möglich ist, geometrische Konstruktionen durchzuführen, und das einen Zugmodus hat. Im Zugmodus können geometrische Objekte, die mindesten einen Freiheitsgrad haben, mit der Maus bewegt werden. Dabei paßt sich die gesamte geometrische Konstruktion der Bewegung an, indem der Computer das entstehende Pfadverfolgungsproblem löst. In dem von uns verwendeten Modell für dynamische Geometrie steht die Stetigkeit der resultierenden Bewegungen im Vordergrund, es wurde von Kortenkamp und Richter- Gebert entwickelt und ist die Grundlage für die Geometriesoftware Cinderella. Wir arbeiten den Zusammenhang dieses Modells zu Riemannschen Flächen algebraischer Funktionen heraus. Im Rahmen dieser Doktorarbeit zeigen wir, wie sich eine algebraische Variante des Modells für Dynamische Geometrie von Kortenkamp und Richter-Gebert sowohl in die angewandte als auch in die reine Mathematik einfügt. Daraus resultiert ein numerisches Verfahren für das Tracing Problem, das auf einer allgemeinen Prediktor-Korrektor-Methode aufbaut. Wie bei den meisten numerischen Verfahren gibt es hierbei keine Garantie dafür, dass die Schrittweite klein genug gewählt ist, um auf dem richtigen Lösungsweg zu bleiben. Das bedeutet, dass ein korrekter Umgang mit Mehrdeutigkeiten nicht garantiert werden kann. Wir haben einen weiteren Algorithmus entwickelt, bei dem die Schrittweite mit Hilfe von Intervallrechnung so gew\"ahlt wird, dass die Korrektheit der Lösung garantiert ist. Kritische Punkte werden durch Umwege umgangen, bei denen die geometrischen Objekte bzw.~die entsprechenden Variablen in einem algebraischen Modell komplexe Koordinaten haben können. Dabei hängt die erreichte Konfiguration wesentlich von dem gewählten Umweg ab. Diese Idee von Kortenkamp und Richter-Gebert führt zu einer konsistenten Behandlung von kritischen Punkten und kommt in der interaktiven Geometriesoftware Cinderella zum Einsatz

Adaptive Sampling for Geometric Approximation

Geometric approximation of multi-dimensional data sets is an essential algorithmic component for applications in machine learning, computer graphics, and scientific computing. This dissertation promotes an algorithmic sampling methodology for a number of fundamental approximation problems in computational geometry. For each problem, the proposed sampling technique is carefully adapted to the geometry of the input data and the functions to be approximated. In particular, we study proximity queries in spaces of constant dimension and mesh generation in 3D. We start with polytope membership queries, where query points are tested for inclusion in a convex polytope. Trading-off accuracy for efficiency, we tolerate one-sided errors for points within an epsilon-expansion of the polytope. We propose a sampling strategy for the placement of covering ellipsoids sensitive to the local shape of the polytope. The key insight is to realize the samples as Delone sets in the intrinsic Hilbert metric. Using this intrinsic formulation, we considerably simplify state-of-the-art techniques yielding an intuitive and optimal data structure. Next, we study nearest-neighbor queries which retrieve the most similar data point to a given query point. To accommodate more general measures of similarity, we consider non-Euclidean distances including convex distance functions and Bregman divergences. Again, we tolerate multiplicative errors retrieving any point no farther than (1+epsilon) times the distance to the nearest neighbor. We propose a sampling strategy sensitive to the local distribution of points and the gradient of the distance functions. Combined with a careful regularization of the distance minimizers, we obtain a generalized data structure that essentially matches state-of-the-art results specific to the Euclidean distance. Finally, we investigate the generation of Voronoi meshes, where a given domain is decomposed into Voronoi cells as desired for a number of important solvers in computational fluid dynamics. The challenge is to arrange the cells near the boundary to yield an accurate surface approximation without sacrificing quality. We propose a sampling algorithm for the placement of seeds to induce a boundary-conforming Voronoi mesh of the correct topology, with a careful treatment of sharp and non-manifold features. The proposed algorithm achieves significant quality improvements over state-of-the-art polyhedral meshing based on clipped Voronoi cells

AutoGraff: towards a computational understanding of graffiti writing and related art forms.

The aim of this thesis is to develop a system that generates letters and pictures with a style that is immediately recognizable as graffiti art or calligraphy. The proposed system can be used similarly to, and in tight integration with, conventional computer-aided geometric design tools and can be used to generate synthetic graffiti content for urban environments in games and in movies, and to guide robotic or fabrication systems that can materialise the output of the system with physical drawing media. The thesis is divided into two main parts. The first part describes a set of stroke primitives, building blocks that can be combined to generate different designs that resemble graffiti or calligraphy. These primitives mimic the process typically used to design graffiti letters and exploit well known principles of motor control to model the way in which an artist moves when incrementally tracing stylised letter forms. The second part demonstrates how these stroke primitives can be automatically recovered from input geometry defined in vector form, such as the digitised traces of writing made by a user, or the glyph outlines in a font. This procedure converts the input geometry into a seed that can be transformed into a variety of calligraphic and graffiti stylisations, which depend on parametric variations of the strokes