Volumetric particle modeling

This dissertation presents a robust method of modeling objects and forces for computer animation. Within this method objects and forces are represented as particles. As in most modeling systems, the movement of objects is driven by physically based forces. The usage of particles, however, allows more artistically motivated behavior to be achieved and also allows the modeling of heterogeneous objects and objects in different state phases: solid, liquid or gas. By using invisible particles to propagate forces through the modeling environment complex behavior is achieved through the interaction of relatively simple components. In sum, 'macroscopic' behavior emerges from 'microscopic' modeling. We present a newly developed modeling framework expanding on related work. This framework allows objects and forces to be modeled using particle representations and provides the details on how objects are created, how they interact, and how they may be displayed. We present examples to demonstrate the viability and robustness of the developed method of modeling. They illustrate the breaking and fracturing of solids, the interaction of objects in different phase states, and the achievement of a reasonable balance between artistic and physically based behaviors

Hierarchical processing, editing and rendering of acquired geometry

La représentation des surfaces du monde réel dans la mémoire d’une machine peut désormais être obtenue automatiquement via divers périphériques de capture tels que les scanners 3D. Ces nouvelles sources de données, précises et rapides, amplifient de plusieurs ordres de grandeur la résolution des surfaces 3D, apportant un niveau de précision élevé pour les applications nécessitant des modèles numériques de surfaces telles que la conception assistée par ordinateur, la simulation physique, la réalité virtuelle, l’imagerie médicale, l’architecture, l’étude archéologique, les effets spéciaux, l’animation ou bien encore les jeux video. Malheureusement, la richesse de la géométrie produite par ces méthodes induit une grande, voire gigantesque masse de données à traiter, nécessitant de nouvelles structures de données et de nouveaux algorithmes capables de passer à l’échelle d’objets pouvant atteindre le milliard d’échantillons. Dans cette thèse, je propose des solutions performantes en temps et en espace aux problèmes de la modélisation, du traitement géométrique, de l’édition intéractive et de la visualisation de ces surfaces 3D complexes. La méthodologie adoptée pendant l’élaboration transverse de ces nouveaux algorithmes est articulée autour de 4 éléments clés : une approche hiérarchique systématique, une réduction locale de la dimension des problèmes, un principe d’échantillonage-reconstruction et une indépendance à l’énumération explicite des relations topologiques aussi appelée approche basée-points. En pratique, ce manuscrit propose un certain nombre de contributions, parmi lesquelles : une nouvelle structure hiérarchique hybride de partitionnement, l’Arbre Volume-Surface (VS-Tree) ainsi que de nouveaux algorithmes de simplification et de reconstruction ; un système d’édition intéractive de grands objets ; un noyau temps-réel de synthèse géométrique par raffinement et une structure multi-résolution offrant un rendu efficace de grands objets. Ces structures, algorithmes et systèmes forment une chaîne capable de traiter les objets en provenance du pipeline d’acquisition, qu’ils soient représentés par des nuages de points ou des maillages, possiblement non 2-variétés. Les solutions obtenues ont été appliquées avec succès aux données issues des divers domaines d’application précités.Digital representations of real-world surfaces can now be obtained automatically using various acquisition devices such as 3D scanners and stereo camera systems. These new fast and accurate data sources increase 3D surface resolution by several orders of magnitude, borrowing higher precision to applications which require digital surfaces. All major computer graphics applications can take benefit of this automatic modeling process, including: computer-aided design, physical simulation, virtual reality, medical imaging, architecture, archaeological study, special effects, computer animation and video games. Unfortunately, the richness of the geometry produced by these media comes at the price of a large, possibility gigantic, amount of data which requires new efficient data structures and algorithms offering scalability for processing such objects. This thesis proposes time and space efficient solutions for modeling, editing and rendering such complex surfaces, solving these problems with new algorithms sharing 4 fundamental elements: a systematic hierarchical approach, a local dimension reduction, a sampling-reconstruction paradigm and a pointbased basis. Basically, this manuscript proposes several contributions, including: a new hierarchical space subdivision structure, the Volume-Surface Tree, for geometry processing such as simplification and reconstruction; a streaming system featuring new algorithms for interactive editing of large objects, an appearancepreserving multiresolution structure for efficient rendering of large point-based surfaces, and a generic kernel for real-time geometry synthesis by refinement. These elements form a pipeline able to process acquired geometry, either represented by point clouds or non-manifold meshes. Effective results have been successfully obtained with data coming from the various applications mentioned

Meshless Mechanics and Point-Based Visualization Methods for Surgical Simulations

Computer-based modeling and simulation practices have become an integral part of the medical education field. For surgical simulation applications, realistic constitutive modeling of soft tissue is considered to be one of the most challenging aspects of the problem, because biomechanical soft-tissue models need to reflect the correct elastic response, have to be efficient in order to run at interactive simulation rates, and be able to support operations such as cuts and sutures. Mesh-based solutions, where the connections between the individual degrees of freedom (DoF) are defined explicitly, have been the traditional choice to approach these problems. However, when the problem under investigation contains a discontinuity that disrupts the connectivity between the DoFs, the underlying mesh structure has to be reconfigured in order to handle the newly introduced discontinuity correctly. This reconfiguration for mesh-based techniques is typically called dynamic remeshing, and most of the time it causes the performance bottleneck in the simulation. In this dissertation, the efficiency of point-based meshless methods is investigated for both constitutive modeling of elastic soft tissues and visualization of simulation objects, where arbitrary discontinuities/cuts are applied to the objects in the context of surgical simulation. The point-based deformable object modeling problem is examined in three functional aspects: modeling continuous elastic deformations with, handling discontinuities in, and visualizing a point-based object. Algorithmic and implementation details of the presented techniques are discussed in the dissertation. The presented point-based techniques are implemented as separate components and integrated into the open-source software framework SOFA. The presented meshless continuum mechanics model of elastic tissue were verified by comparing it to the Hertzian non-adhesive frictionless contact theory. Virtual experiments were setup with a point-based deformable block and a rigid indenter, and force-displacement curves obtained from the virtual experiments were compared to the theoretical solutions. The meshless mechanics model of soft tissue and the integrated novel discontinuity treatment technique discussed in this dissertation allows handling cuts of arbitrary shape. The implemented enrichment technique not only modifies the internal mechanics of the soft tissue model, but also updates the point-based visual representation in an efficient way preventing the use of costly dynamic remeshing operations

Point based graphics rendering with unified scalability solutions.

Standard real-time 3D graphics rendering algorithms use brute force polygon rendering, with complexity linear in the number of polygons and little regard for limiting processing to data that contributes to the image. Modern hardware can now render smaller scenes to pixel levels of detail, relaxing surface connectivity requirements. Sub-linear scalability optimizations are typically self-contained, requiring specific data structures, without shared functions and data. A new point based rendering algorithm 'Canopy' is investigated that combines multiple typically sub-linear scalability solutions, using a small core of data structures. Specifically, locale management, hierarchical view volume culling, backface culling, occlusion culling, level of detail and depth ordering are addressed. To demonstrate versatility further, shadows and collision detection are examined. Polygon models are voxelized with interpolated attributes to provide points. A scene tree is constructed, based on a BSP tree of points, with compressed attributes. The scene tree is embedded in a compressed, partitioned, procedurally based scene graph architecture that mimics conventional systems with groups, instancing, inlines and basic read on demand rendering from backing store. Hierarchical scene tree refinement constructs an image tree image space equivalent, with object space scene node points projected, forming image node equivalents. An image graph of image nodes is maintained, describing image and object space occlusion relationships, hierarchically refined with front to back ordering to a specified threshold whilst occlusion culling with occluder fusion. Visible nodes at medium levels of detail are refined further to rasterization scales. Occlusion culling defines a set of visible nodes that can support caching for temporal coherence. Occlusion culling is approximate, possibly not suiting critical applications. Qualities and performance are tested against standard rendering. Although the algorithm has a 0(f) upper bound in the scene sizef, it is shown to practically scale sub-linearly. Scenes with several hundred billion polygons conventionally, are rendered at interactive frame rates with minimal graphics hardware support

Subdivision Surface based One-Piece Representation

Subdivision surfaces are capable of modeling and representing complex shapes of arbi-trary topology. However, methods on how to build the control mesh of a complex surfaceare not studied much. Currently, most meshes of complicated objects come from trian-gulation and simplification of raster scanned data points, like the Stanford 3D ScanningRepository. This approach is costly and leads to very dense meshes.Subdivision surface based one-piece representation means to represent the final objectin a design process with only one subdivision surface, no matter how complicated theobject\u27s topology or shape. Hence the number of parts in the final representation isalways one.In this dissertation we present necessary mathematical theories and geometric algo-rithms to support subdivision surface based one-piece representation. First, an explicitparametrization method is presented for exact evaluation of Catmull-Clark subdivisionsurfaces. Based on it, two approaches are proposed for constructing the one-piece rep-resentation of a given object with arbitrary topology. One approach is to construct theone-piece representation by using the interpolation technique. Interpolation is a naturalway to build models, but the fairness of the interpolating surface is a big concern inprevious methods. With similarity based interpolation technique, we can obtain bet-ter modeling results with less undesired artifacts and undulations. Another approachis through performing Boolean operations. Up to this point, accurate Boolean oper-ations over subdivision surfaces are not approached yet in the literature. We presenta robust and error controllable Boolean operation method which results in a one-piecerepresentation. Because one-piece representations resulting from the above two methodsare usually dense, error controllable simplification of one-piece representations is needed.Two methods are presented for this purpose: adaptive tessellation and multiresolutionanalysis. Both methods can significantly reduce the complexity of a one-piece represen-tation and while having accurate error estimation.A system that performs subdivision surface based one-piece representation was im-plemented and a lot of examples have been tested. All the examples show that our ap-proaches can obtain very good subdivision based one-piece representation results. Eventhough our methods are based on Catmull-Clark subdivision scheme, we believe they canbe adapted to other subdivision schemes as well with small modifications

An image-space algorithm for hardware-based rendering of constructive solid geometry

A new approach to image-space hardware-based rendering of Constructive Solid Geometry (CSG) models is presented. The work is motivated by the evolving functionality and performance of computer graphics hardware. This work is also motivated by a specific industrial application --- interactive verification of five axis grinding machine tool programs. The goal is to minimise the amount of time required to render each frame in an animation or interactive application involving boolean combinations of three dimensional shapes. The Sequenced Convex Subtraction (SCS) algorithm utilises sequenced subtraction of convex objects for the purpose of interactive CSG rendering. Concave shapes must be decomposed into convex shapes for the purpose of rendering. The length of Permutation Embedding Sequences (PESs) used as subtraction sequences are shown to have a quadratic lower bound. In ma ny situations shorter sequences can be used, in the best case linear. Approaches to subtraction sequence encoding are presented including the use of object-space overlap information. The implementation of the algorithm is experimentally shown to perform better on modern commodity graphics hardware than previously reported methods. This work also examines performance aspects of the SCS algorithm itself. Overall performance depends on hardware characteristics, the number and spatial arrangement of primitives, and the structure and boolean operators of the CSG tree

Courbure discrète : théorie et applications

International audienceThe present volume contains the proceedings of the 2013 Meeting on discrete curvature, held at CIRM, Luminy, France. The aim of this meeting was to bring together researchers from various backgrounds, ranging from mathematics to computer science, with a focus on both theory and applications. With 27 invited talks and 8 posters, the conference attracted 70 researchers from all over the world. The challenge of finding a common ground on the topic of discrete curvature was met with success, and these proceedings are a testimony of this wor

Frictional Contact in Interactive Deformable Environments

L\u2019uso di simulazioni garantisce notevoli vantaggi in termini di economia, realismo e di flessibilit\ue0 in molte aree di ricerca e in ambito dello sviluppo tecnologico. Per questo motivo le simulazioni vengono usate spesso in ambiti quali la prototipazione di parti meccaniche, nella pianificazione e nell\u2019addestramento di procedure di assemblaggio e disassemblaggio inoltre, di recente, le simulazioni si sono dimostrate validi strumenti anche nell\u2019assistenza e nell\u2019addestramento ai chirurghi, in particolare nel caso della chirurgia laparoscopica. La chirurgia laparoscopica, infatti, \ue8 considerata lo standard per molte procedure chirurgiche. La principale differenza rispetto alla chirurgia tradizionale risiede nella notevole limitazione che ha il chirurgo nell\u2019interagire e nel percepire l\u2019ambiente in lavora, sia nella vista che nel tatto. Questo rappresenta una forte limitazione per il chirurgo a cui \ue8 richiesta una lunga fase di addestramento prima di poter ottenere la necessaria destrezza per intervenire in laparoscopia con profitto. Queste limitazioni, d\u2019altra parte, rendono la laparoscopia il candidato ideale per l\u2019introduzione della simulazione nell\u2019addestramento. Attualmente sono disponibili in commercio dei software per l\u2019addestramento alla laparoscopia, tuttavia essi sono in genere basati su modelli rigidi, o modelli che comunque mancano del necessario realismo fisico. L\u2019introduzione di modelli deformabili migliorerebbe notevolmente l\u2019accuratezza e il realismo delle simulazioni. Nel caso dell\u2019addestramento il maggior realismo permetterebbe all\u2019utente di acquisire non solo le conoscenze motorie basilari ma anche capacit\ue0 e conoscenze di pi\uf9 alto livello. I corpi rigidi, infatti, rappresentano una buona approssimazione della realt\ue0 solo in situazioni particolari ed entro intervalli di sollecitazioni molto ristretti. Quando si considerano materiali non ingegneristici, come accade nelle simulazioni chirurgiche, le deformazioni non possono essere trascurate senza compromettere irrimediabilmente il realismo dei risultati. L\u2019uso di modelli deformabili tuttavia introduce notevole complessit\ue0 computazionale per il calcolo della fisica che regola le deformazioni e limita fortemente l\u2019uso di dati precalcolati, spesso utilizzati per velocizzare la fase di identificazione delle collisioni tra i corpi. I ritardi dovuti all\u2019uso di modelli deformabili rappresentano un grosso limite soprattutto nelle applicazioni interattive che, per consentire all\u2019utente di interagire con l\u2019ambiente, richiedono il calcolo della simulazione entro intervalli di tempo molto ridotti. In questa tesi viene affrontato il tema della simulazione di ambienti interattivi composti da corpi deformabili che interagiscono con attrito. Vengono analizzati e sviluppati differenti tecniche e metodi per le diverse componenti della simulazione: dalla simulazione di modelli deformabili, agli algoritmi di identificazione e soluzione delle collisioni e alla modellazione e integrazione dell\u2019attrito nella simulazione. In particolare vengono valutati i principali metodi che rappresentano lo stato dell\u2019arte nella modellazione di materiali deformabili. L\u2019analisi considera i fondamenti fisici su cui i modelli si basano e quindi sul grado di realismo che possono garantire in termini di deformazioni modellabili e la semplicit\ue0 d\u2019uso degli stessi (ovvero la facilit\ue0 di comprensione del metodo, la calibrazione del modello e la possibilit\ue0 di adattare il modello a situazioni differenti) ma viene considerata anche la complessit\ue0 computazionale di ciascun metodo in quanto essa rappresenta un fattore estremamente importante nella scelta e nell\u2019uso dei modelli deformabili nelle simulazioni. Il confronto dei differenti modelli e le caratteristiche identificate hanno motivato lo sviluppo di un metodo innovativo per fornire un\u2019interfaccia comune ai vari metodi di simulazione dei tessuti deformabili. Tale interfaccia ha il vantaggio di fornire dei metodi omogenei per la manipolazione dei diversi modelli deformabili. Ci\uf2 garantisce la possibilit\ue0 di scambiare il modello usato per la simulazione delle deformazioni mantenendo inalterati le altre strutture dati e i metodi della simulazione. L\u2019introduzione di tale interfaccia unificata si dimostra particolarmente vantaggiosa in quanto permette l\u2019uso di un solo metodo per l\u2019identificazione delle collisioni per tutti i differenti modelli deformabili. Ci\uf2 semplifica molto l\u2019analisi e la definizione dei requisiti di tale modulo software. L\u2019identificazione delle collisioni tra modelli rigidi generalmente precalcola delle partizioni dello spazio in cui i corpi sono definiti oppure sfrutta la suddivisione del corpo analizzato in parti convesse per velocizzare la simulazione. Nel caso di modelli deformabili non \ue8 possibile applicare tali tecniche a causa dei continui cambiamenti nella configurazione dei corpi. Dopo che le collisioni tra i corpi sono state riconosciute e che i punti di contatto sono stati identificati e necessario risolvere le collisioni tenendo conto della fisica sottostante i contatti. Per garantire il realismo \ue8 necessario assicurare che i corpi non si compenetrino mai e che nella simulazione delle collisioni tutti i fenomeni fisici di interesse coinvolti nel contatto tra i corpi vengano considerati: questi includono le forze elastiche che si esercitano tra i corpi e le forze di attrito che si generano lungo le superfici di contatto. L\u2019innovativo metodo proposto per la soluzione delle collisioni garantisce il realismo della simulazione e l\u2019integrazione con l\u2019interfaccia proposta per la gestione unificata dei modelli. Una caratteristica importante dei tessuti biologici \ue8 il comportamento anisotropico, dovuto, in genere, alla loro struttura fibrosa. In questa tesi viene proposto un nuovo metodo per aggiungere l\u2019anisotropia al comportamento dei modelli massa molla. Il metodo ha il vantaggio di mantenere la velocit\ue0 computazionale e la semplicit\ue0 di implementazione dei modelli massa molla classici e riesce a differenziare efficacemente la risposta del modello alle sollecitazioni lungo le differenti direzioni. Le tecniche descritte sono state integrate in due applicazioni che forniscono la simulazione della fisica di ambienti con corpi deformabili. La prima delle due implementa tutti i metodi descritti per la simulazione dei modelli deformabili, identifica le collisioni con precisione e le risolve fornendo la possibilit\ue0 di scegliere il modello di attrito pi\uf9 adatto, dimostrando cos\uec la fattibilit\ue0 dell\u2019approccio proposto. La limitazione principale di tale simulatore risiede nell\u2019alto tempo di calcolo richiesto per la simulazione dei singoli passi di simulazione. Tale limitazione \ue8 stata superata in una seconda implementazione che sfrutta il parallelismo intrinseco delle simulazioni fisiche per ottimizzare gli algoritmi e che, quindi, riesce a sfruttare al meglio la potenza computazionale delle architetture hardware parallele. Al fine di ottenere le prestazioni richieste per la simulazione di ambienti interattivi con ritorno di forza, la simulazione \ue8 basata su un algoritmo di identificazione delle collisioni semplificato, ma implementa gli altri metodi descritti in questa tesi. L\u2019implementazione parallela sfrutta le capacit\ue0 di calcolo delle moderne schede video munite di processori altamente paralleli e ci\uf2 permette di aggiornare la scena ogni millisecondo. Questo elimina ogni discontinuit\ue0 nel ritorno di forza reso all\u2019utente e nell\u2019aggiornamento della grafica della scena, inoltre garantisce il realismo necessario alla simulazione fisica sottostante. Le applicazioni implementate provano la fattibilit\ue0 della simulazione della fisica di interazioni complesse tra corpi deformabili. Inoltre, l\u2019implementazione parallela della simulazione rappresenta un promettente punto di partenza per la realizzazione di simulazioni interattive che potr\ue0 essere utilizzato in ambiti di ricerca differenti, quali l\u2019addestramento di chirurghi o la prototipazione rapida.The use of simulations provides great advantages in term of economy, realism, and adaptability to user requirements in many research and technological fields. For this reason simulations are currently exploited, for example, in prototyping of machinery parts, in assembly-disassembly test or training and, recently, simulations have also allowed the development of many useful and promising tools for the assistance and learning of surgical procedures. This is particularly true for laparoscopic intervention. Laparoscopy, in fact, represents the gold standard for many surgical procedures. The principal difference from standard surgery is the reduction of the surgeon ability to perceive the surgical scenario, both from visual and tactile point of view. This represents a great limitation for surgeons who undergo long training before being able to perform laparoscopic intervention with proficiency. This, on the other hand, makes laparoscopy an excellent candidate for the use of simulations for training. Some commercial training softwares are already available on the market, but they are usually based on rigid body models that completely lack the physical realism. The introduction of deformable models may leads to a great increment in terms of realism and accuracy. And, in the case of laparoscopy trainer it may allow the user to learn not only basic motor skills, but also higher level capabilities and knowledge. Rigid bodies, in fact, represents a good approximation of reality only in some situations and in very restricted ranges of solicitations. In particular, when non engineering materials are involved, as happens in surgical simulations, deformations cannot be neglected without completely loosing the realism of the environment. The use of deformable models, however, is limited for the high computational costs involved in the computation of the physics undergoing the deformations and because of the reduction in pre computable data in particular for collision detection between bodies. This represents a very limiting factor in interactive environments where, to allow the user to interactively control the virtual bodies, the simulation should be performed in real time. In this thesis we address the simulation of interactive environment populated with deformable models that interact with frictional contacts. This includes the analysis and the development of different techniques which implement the various parts of the simulation: mainly the methods for the simulation of deformable models, the collision detection and collision solution techniques but also the modelling and the integration of suitable friction models in the simulation. In particular we evaluated the principal methods that represent the state of the art in soft tissue modeling. Our analysis is based on the physical background of each method and thus on its realism in terms of deformations that the method can mimic and on the ease of use (i.e. method understanding, calibration and ability to adapt to different scenarios) but we also compared the computational complexity of different models, as it represents an extremely important factor in the choice and in the use of models in simulations. The comparison of different features in analyzed methods motivated us to the development of an innovative method to wrap in a common representation framework different methodologies of soft tissue simulation. This framework has the advantage of providing a unified interface for all the deformable models and thus it provides the ability to switch between deformable model keeping unchanged all other data structures and methods of the simulation. The use of this unique interface allows us to use one single method to perform the collision detection phase for all the analyzed deformable models, this greatly helped during the identification of requirements and features of such software module. Collision detection phase, when applied to rigid bodies, usually takes advantage of pre computation to subdivide body shapes in convex elements or to construct partitions of the space in which the body is defined to speed up the computation. When handling deformable models this is not possible because of the continuous changes in bodies shape. The collision detection method used in this work takes into account this problem and regularly adapt the data structures to the body configuration. After collisions have been detected and contact points have been identified on colliding bodies, it is necessary to solve the collision in a physics based way. To this extent we have to ensure that objects never compenetrate during the simulation and that, when solving collisions, all the physical phenomena involved in the contact of real bodies are taken into account: this include the elastic response of bodies during the contact and the frictional force exerted between each pair of colliding bodies. The innovative method for solving collision that we describe in this thesis ensures the realism of the simulation and the seamless interaction with the common framework used to integrate deformable models. One important feature of biologic tissues is their anisotropic behavior that usually comes from the fibrous structure of these tissues. In this thesis we propose a new method to introduce anisotropy in mass spring model. The method has the advantages of preserving the speed and ease of implementation of the model and it effectively introduces differentiation of the model behavior along the chosen directions. The described techniques have been integrated in two applications that allows the physical simulation of environments populated with deformable models. The first application implements all the described methods to simulate deformable models, it performs precise collision detection and solution with the possibility to chose the most suitable friction model for the simulation. It demonstrates the effectiveness of the proposed framework. The main limitation of this simulator, i.e. its high computation time, is tackled and solved in a second application that exploits the intrinsic parallelism of physical simulations to optimize the implementation and to exploit parallel architecture computational power. To obtain the performances required for an interactive environment the simulation is based on a simplified collision detection algorithm, but it features all the other techniques described in this thesis. The parallel implementation exploits graphic cards processor, a highly parallel architecture that update the scene every milliseconds. This allows the rendering of smooth haptic feedback to the user and ensures the realism of the physics simulation. The implemented applications prove the feasibility of the simulation of complex interactions between deformable models with physics realism. In addition, the parallel implementation of the simulator represents a promising starting point for the development of interactive simulations that can be used in different fields of research, such as surgeon training or fast prototyping

Boolean Operations on Surfel-Bounded Objects using Constrained BSP-Trees

Point-based modeling and rendering is an active area of research in Computer Graphics. The concept of points with attributes (e.g. normals) is usually referred to as surfels, and many algorithms have been devised to their efficient manipulation and rendering. Key to the efficiency of many methods is the use of partitioning schemes, and usually axis-aligned structures such as octrees and KDtrees are preferred, instead of more general BSP-trees. In this work we introduce a data structure called Constrained BSP-tree (CBSP-tree) that can be seen as an intermediate structure between KD-trees and BSP-trees. The CBSPtree is characterized by allowing arbitrary cuts as long as the complexity of its cells remains bounded, allowing better approximation of curved regions. We discuss algorithms to build CBSP-trees using the flexibility that the structure offers, and present a modified algorithm for boolean operations that uses a new inside-outside object classification. Results show that CBSP-trees generate fewer cells than axis-aligned structures. 1

Génération et édition de textures géométriques représentées par des ensembles de points

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