Détection d’intersection via l’application de Gauss; revue et nouvelles techniques

This paper delves into the problem of detecting the intersection of two convexpolyhedra.It does so through the lens of Minkowski sums and Gauss maps, and with a biastowards applications in computer graphics and robotics.In the first part, we show how Minkowski sums and Gauss maps come into play,give a brief survey of techniques for pairs of simple shapes and describe alow-level optimization of a naive algorithm for convex polyhedra, which isapplied to tetrahedra.Novel applications to the ray casting problem are also given.In the second part, we take a more abstract approach to the problem anddescribe a new and very efficient and robust algorithm for detecting theintersection of two convex shapes.The new technique works directly on the unit sphere, interpreted as the sphereof directions.In particular, it is compared favourably to the ubiquitous algorithm ofGilbert, Johnson and Keerthi.Cet article discute du problème (décisionnel) de la détection de l'intersectionde deux polyèdres convexes. Il porte particulièrement sur les applications dece problème en informatique graphique et en robotique. La discussion s'y faitdu point de vue des sommes de Minkoswki et de l'application de Gauss.Dans la première partie, nous rappellons le rôle de ce point de vue dans lacompréhension de la géométrie du problème. Nous donnons un bref aperçu destechniques conçues pour certaines paires de formes simples, et nous proposonsun algorithme naïf mais optimisé, traitant des polyèdres convexes quelconques.Nous traitons en exemple une application aux paires de tétraèdres et uneapplication au problème du lancer de rayons.En deuxième partie, nous approchons le problème de manière plus abstraite etdécrivons un nouvel algorithme robuste et rapide pour la détection del'intersection de deux objets convexes (non nécessairement polyédrique).Ce nouvel algorithme travaille directement sur la sphère unité que nousinterprétons comme l'espace des directions. En particulier, notre nouvelletechnique est comparée favorablement à celle, fort répandue, de Gilbert,Johnson et Keerthi

Detecting the intersection of two convex shapes by searching on the 2-sphere

Special issue: Proceedings of Solid and Physical Modeling (SPM 2017)International audienceWe take a look at the problem of deciding whether two convex shapes intersect or not. We do so through the well known lens of Minkowski sums and with a bias towards applications in computer graphics and robotics. We describe a new technique that works explicitly on the unit sphere, interpreted as the sphere of directions. In extensive benchmarks against various well-known techniques, ours is found to be slightly more efficient, much more robust and comparatively easy to implement. In particular, our technique is compared favourably to the ubiquitous algorithm of Gilbert, Johnson and Keerthi (GJK), and its decision variant by Gilbert and Foo. We provide an in-depth geometrical understanding of the differences between GJK and our technique and conclude that our technique is probably a good drop-in replacement when one is not interested in the actual distance between two non-intersecting shapes

Graphite-MicroMégas, a tool for DNA modeling

National audienceMicroMégas is the current state of an ongoing effort to develop tools for modeling biological assembly of molecules. We here present its DNA modeling part. MicroMégas is implemented as a plug-in to Graphite, which is a research plat- form for computer graphics, 3D modeling and numerical geometry that is developed by members of the ALICE team of INRIA.Nous décrivons l'outils MicroMégas et les techniques qu'il met en jeu pour la modélisation d'assemblage de molécule, en particulier la modélisation de brin d'ADN

Iterative carving for self-supporting 3D printed cavities

International audienceAdditive manufacturing technologies fabricate objects layer by layer, adding material on top of already solidified layers. A key challenge is to ensure that there is always material below, for otherwise added material simply falls under the effect of gravity. This is a critical issue with most technologies, and with fused filament in particular. In this work we investigate how to compute as large as possible empty cavities which boundaries are self-supporting. Our technique is based on an iterated carving algorithm, that is fast to compute and produces nested sets of inner walls. The walls have exactly the minimal printable thickness of the manufacturing process everywhere. Remarkably, our technique is out-of-core, sweeping through the model from the top down. Using our approach, we can print large objects using as little as a single filament thickness for the boundary, providing one order of magnitude reduction in print time and material use while guaranteeing printability

Farthest-Polygon Voronoi Diagrams

Given a family of k disjoint connected polygonal sites in general position and of total complexity n, we consider the farthest-site Voronoi diagram of these sites, where the distance to a site is the distance to a closest point on it. We show that the complexity of this diagram is O(n), and give an O(n log^3 n) time algorithm to compute it. We also prove a number of structural properties of this diagram. In particular, a Voronoi region may consist of k-1 connected components, but if one component is bounded, then it is equal to the entire region

HA-Buffer: Coherent Hashing for single-pass A-buffer

Identifying all the surfaces projecting into a pixel has several important applications in Computer Graphics, such as transparency and CSG. These applications further require ordering, in each pixel, the surfaces by their distance to the viewer. In real-time rendering engines, this is often achieved by recording sorted lists of the fragments produced by the rasterization pipeline. The major challenge is that the number of fragments is not known in advance. This results in computational and memory overheads due to the necessary dynamic nature of the data-structure. Similarly, many fragments which are not useful for the final image--due to opacity accumulation for instance--have to be stored and sorted nonetheless, negatively impacting performance. This paper proposes a novel approach which records and simultaneously sorts all fragments in a single geometry pass. The storage overhead per fragment is typically lower than 8 bits per record, and no pointers are involved. Since fragments are progressively sorted in memory, it is possible to assess during rendering whether a new fragment is useful. Our approach combines advantages of previous approaches at similar levels of performance, and is implemented in a single fragment shader of 24 lines of GLSL.Plusieurs applications en synthèse d'image nécessitent le calcul de l'ensemble des surfaces visibles au travers d'un pixel. Citons le dessin correct de surfaces transparentes ainsi que le dessin de mod'eles CSG. Ces applications nécessite également de trier les surfaces, pour chaque pixel, selon leur distance au point de vue. Pour les applications en temps-réel, ce sont les fragments produits par l'étape de rasterisation qui sont triés et stockés en mémoire vidéo. Le nombre de ces fragments n'étant pas connu à l'avance, il est nécessaire d'utiliser de coûteuses techniques de gestion de la mémoire. De plus, tous les fragments sont traités même si une fraction non négligeable d'entre eux peut être inutile au dessin de l'image finale (grâce, par exemple, à l'accumulation de l'opacité de plusieurs surfaces combinées). Nous proposons une technique simple pour trier les fragments d'un même pixel au moment de leur rasterisation, sans utiliser de liste chainée (et donc de pointeur). Puisque la liste des fragments pour un pixel est toujours triée, il est possible de déterminer, au moment de sa rasterisation, si un fragment contribuera ou pas à l'image finale, et de le rejetter le cas échéant. La technique combine les avantages de plusieurs approches existantes pour un niveau de performance similaire. Elle a l'unique avantage d'étre très simple à coder : 24 lignes de GLSL

Radiance Space, as Represented by the Visibility 2-Skeleton

International audienceSince radiance in a vacuum remains constant along rays, the radiance in a scene's free space can be represented as a real function over the space of maximal segments. This space is four dimensional and orga- nized by the visibility complex. To date only the 0-D and 1-D cells of the visibility complex have been constructed and implemented. In order to use this topological space to support radiance samples for image-based rendering, form-factor computations, global illumination and shadowing, we also need to construct its 2-D cells. This paper expands the frontier of our understanding of the visibility complex by constructing and im- plementing its 2-D cells. We demonstrate the capabilities of these 2-cells by using them to organize sparse illumination samples in a scene. These samples interpolate only within the boundaries of the 2-cells, thus avoid- ing aliasing while respecting sharp changes in illumination

Implicit Modelling Using Subdivision-curves

International audienceTo remain an attractive model, skeleton-based implicit surfaces have to allow the design and display of shapes at interactive rates. This paper focuses on surfaces whose skeletons are graphs of interconnected curves. We present subdivision-curve primitives that rely on convolution for generating bulge-free and crease-free implicit surfaces. These surfaces are efficiently yet correctly displayed using local meshes around each curve that locally overlap in blending regions. Subdivision-curve primitives offer a practical solution to the unwanted-blending problem that ensures $C^1$ continuity everywhere. Moreover, they can be used to generate representations at different levels of detail, enabling the interactive display of at least a coarse version of the objects, whatever the performance of the workstation. We also present a practical solution to the unwanted blending problem, used to avoid blending between parts of the surface that do not correspond to neighbouring skeletal elements

Chained segment offsetting for ray-based solid representations

International audienceWe present a novel approach to offset solids in the context of fabrication. Our input solids can be given under any representation: boundary meshes, voxels, indicator functions or CSG expressions. The result is a ray-based representation of the offset solid directly used for visualization and fabrication: We never need to recover a boundary mesh in our context. We define the offset solid as a sequence of morphological operations along line segments. This is equivalent to offsetting the surface by a solid defined as a Minkowski sum of segments, also known as a zonotope. A zonotope may be used to approximate the Euclidean ball with precise error bounds. We propose two complementary implementations. The first is dedicated to solids represented by boundary meshes. It performs offsetting by modifying the mesh in sequence. The result is a mesh improper for direct display, but that can be resolved into the correct offset solid through a ray representation. The major advantage of this first approach is that no loss of information – re-sampling – occurs during the offsetting sequence. However, it applies only to boundary meshes and cannot mix sequences of dilations and erosions. Our second implementation is more general as it applies directly to a ray-based representation of any solid and supports any sequence of erosion and dilation along segments. We discuss its fast implementation on modern graphics hardware. Together, the two approaches result in a versatile tool box for the efficient offsetting of solids in the context of fabrication

Polyhedral Voronoi diagrams for additive manufacturing

International audienceA critical advantage of additive manufacturing is its ability to fabricate complex small-scale structures. These microstructures can be understood as a metamaterial: they exist at a much smaller scale than the volume they fill, and are collectively responsible for an average elastic behavior different from that of the base printing material making the fabricated object lighter and/or flexible along specific directions. In addition, the average behavior can be graded spatially by progressively modifying the microstructure geometry.The definition of a microstructure is a careful trade-off between the geometric requirements of manufacturing and the properties one seeks to obtain within a shape: in our case a wide range of elastic behaviors. Most existing microstructures are designed for stereolithography (SLA) and laser sintering (SLS) processes. The requirements are however different than those of continuous deposition systems such as fused filament fabrication (FFF), for which there is currently a lack of microstructures enabling graded elastic behaviors.In this work we introduce a novel type of microstructures that strictly enforce all the requirements of FFF-like processes: continuity, self-support and overhang angles. They offer a range of orthotropic elastic responses that can be graded spatially. This allows to fabricate parts usually reserved to the most advanced technologies on widely available inexpensive printers that also benefit from a continuously expanding range of materials