In this paper , we provide a complete description of dynamic algorithms for constructing convex hulls and Voronoi diagrams of additively weighted points of . The algorithms have been implemented in and experimental results are reported.

Our motivation comes from the fact that weighted points can be considered as hyperspheres when the weights are positive and is twofold. On the one hand, spheres are non linear objects and, besides the combinatorial and algorithmic questions, numerical and robustness issues deserve a careful investigation, which has not been fully done yet. On the other hand, spheres are objects of major concern in various fields, most notably structural biology, and effective implementations of basic geometric algorithms for spheres are needed.

We first revisit the problem of computing the convex hull of nweighted points of . This problem has already been solved optimally. We present a simpler fully dynamic algorithm and provide a complete description of all the predicates for any dimension d.

We apply our result on the construction of the convex hull of additively weighted points to the construction of a Voronoi cell in the Voronoi diagram of nadditively weighted points. The main contribution of this work is to provide a full analysis of the predicates involved, a thorough treatment of the degenerate cases, and a cgalimplementation. Our predicates, when specialized to the planar case ( d= 2), are simpler and of lower degree than the best predicates known so far. Our implementation follows the cgaldesign and, in particular, is made both robust and efficient through the use of a filtered exact arithmetic.

We investigate several problems related to the computation of the visibility complex between three dimensional objects or dynamic updates of the visibility in two dimensions.

A set of objects in three dimensional space is given, we look at the different kind of lines with respect to these objects. With respect to a single object a line can intersect the object, be tangent or miss the object. In general position, a line is tangent to at most four objects, such a line is called freeif it intersects no other objects, it is called free line segmentif it intersects no objects between the points where the line touch the four objects and occludedotherwise.

First, we prove that the lines tangent to four (possibly intersecting) convex polyhedra with nedges in total, form ( n²)connected components in the worst case. In the generic case, each connected component is a single line, but our result still holds for arbitrary degenerate scenes. More generally, we show that a set of kpossibly intersecting convex polyhedra with a total of nedges admits, in the worst case, ( n²k²)connected components of maximal free line segments tangent to any four of the polytopes. This bound also holds for the number of connected components of possibly occluded lines tangent to any four of the polytopes .

In another paper, we investigate the special case of lines tangent to four triangles in three-dimensional space. This give an idea of the constant involved in algorithms on ntriangles where finding such tangents is considered as a constant time problem. We establish a lower bound by giving an example having 62 such tangents. The upper bound on the number of connected components of tangents is 162 for possibly intersecting triangles and decreases to 156 if the triangles are disjoint. In addition, if the triangles are in (algebraic) general position, then the number of tangents is finite and it is always even  .

Finally given a set of npoints in the plane, we consider the problem of computing the circular ordering of the points about a viewpoint qand efficiently maintaining this ordering information as qmoves  . In usual circular ordering an event occur when the viewpoint and two data-point becomes collinear. Here the main difference is that we consider this as an event only if the view point is not in between the two data points. Our main result is that after a O( nlog n)preprocessing, we allow moving point queries along a specified line segment on O( klog n+ s)amortized time where kis the number of different views and sis the output size.

One of the challenge of graph drawing is to draw graphs using a minimal amount of space. One instance of this problem is to allow the graph vertices to lie in 3D at point locations with integer coordinates, and to draw graph edges as non intersectingpolylines with bends allowed at point location with integer coordinates as well. In our work we propose a construction using one bend per edge to draw the complete graph K_ninside a box of volume O( n^2.5). The lower bound for non intersecting 3D drawing of K_nis known to be ( n²), our result represents a significant improvement over previous result of . The result trivially applies to any graph with nvertices that is a subgraph of K_n  , .

Reporting maximal cliques of a graph is a fundamental problem arising in many areas. To the best of our knowledge, this problem is tackled resorting to the old Bron-Kerbosch algorithm (1973), or its recent variant by I. Koch (2001). Both algorithms suffer from a poor output sensitivity and from pathologicalworst-cases.

In this context, this paper makes three contributions. First, we show that a slight modification of the greedy pivoting strategy used by I. Koch allows one to get rid of the pathologicalworst-cases, also improving overall performances. Second, exploiting the recursive structure of non maximal cliques, we show that the pivoting strategy developed by I. Koch is a particular case of a more general optimization strategy based on the concept of dominatednodes. Using different instantiations of this concept, we design four modified Bron-Kerbosch algorithms, with better output-sensitivity. Third, we discuss implementation issues and provide a detailed experimental study on random graphs.

The bottom-line of this study is the investigation of the trade-off between output sensitivity and the overhead associated to the identification of dominated nodes.

Given two graphs, a fundamental task faced by (graph, shape) matching algorithms consists of computing either the (Connected) Maximal Common Induced Subgraphs ((C)MCIS) or the (Connected) Maximal Common Edge Subgraphs ((C)MCES) Consider two graphs F= ( V[ F], E[ F]), G= ( V[ G], E[ G]). A CIS consists of two subsets of vertices , , such that the subgraphs induced by these vertex sets are isomorphic. A CES consists of subsets , , such that and are isomorphic.. In particular, computing the CMCIS or CMCES reduces to reporting so-called c-connected cliquesin product graphs, a problem for which an algorithm has been presented in a recent paper I. Koch, TCS 250 (1-2), 2001. This algorithm suffers from two problems which are corrected in this note .

Work on the cgalcurved kernel has continued, following the past year work on the overall design. We have implemented more primitives dealing with circular arcs, as well as mixed operations with line segments. This allows to compute arrangements of objects commonly used in VLSI (Very Large Scale Integration, see Figure  ). Some contacts with industry (Mania Barco, Dassault Systèmes, GeometryFactory) show that there are needs for robust and efficient computations in this area.

Some optimization work has also been performed on this initial part of the curved kernel. As usual, the computations are done exactly and efficiently through the use of filtering at various stages. Most notably, filtering with bounding boxes and bounding hexagons around circular arcs already shows a positive impact on running time. This is expected to be more important for higher degree curves such as conic arcs. Some extensions to 3D have also been started for handling sphere patches, as well as for benchmarking the existing methods.

Basic geometric primitives are usually classified into predicates, functions which return boolean or enumerated results, and constructions, which return new geometric objects. An example of the former is the orientation predicate, and an example of the latter is the computation of the intersection of two line segments. While very efficient methods are now available to compute exact predicates, geometric constructions are also very important.

We have started implementing a generic mechanism for filtering geometric constructions, and providing a full cgalkernel this way. The 2D version is almost entirely implemented, and it already shows important performance improvements compared to previous solutions based on filtered number types. Many implemented algorithms use geometric constructions and need some guarantees on them, such as the cgalarrangements, Voronoi diagrams and meshes.

We consider the problem of discovering a C²unknown surface Susing tactile probes  . We show that Scan be approximated by a triangulated surface Wwithin any desired accuracy. We also bound the number of probes and the number of elementary moves of the probing device. Our solution is an extension of our previous work on Delaunay refinement techniques for certified surface meshing  . Our approximating surface enjoys the many nice properties of the meshes obtained by those techniques, e.g. exact topological type, facets with bounded aspect ratio.

Similar results were only known for the case of a single planar convex object, a much easier case which reduces to blind approximation of the support function. The non convex case is different in several respects. First, there is no global parametrization of the boundary and, moreover, we cannot simply probe from infinity and need to determine positions outside the object where to place the probing device and to determine paths along which the probing device can be safely moved without colliding with the object.

Following the perception-action-cognition paradigm, we distinguish between the probing cost, the displacement cost and the combinatorial cost. The information cost measures the number of probes. The displacement cost accounts for moving the probing device. The combinatorial cost accounts for the arithmetic operations and comparisons, and the maintenance of the data structures. We show that it is not possible, in general, to simultaneously optimize all costs and analyze them separately.

In the last decade, a great deal of work has been devoted to the elaboration of a sampling theory for smooth surfaces. The goal was to work out sampling conditions that ensure a good reconstruction of a given smooth surface Sfrom a finite subset Eof S. Among these conditions, a prominent one is the -sampling condition of Amenta and Bern, which states that every point pof Sis closer to Ethan lfs ( p), where lfs is the distance from pto the medial axis of S.

Amenta and Bern have proved that it is possible to extract from the Delaunay triangulation of Ea PL surface that approximates Sboth topologically and geometrically (surface reconstruction problem). Last year, we introduced a weaker notion called loose -sample and proposed an algorithm to actually sample and mesh a smooth surface (surface mesh generation) with guarantees.

The sampling conditions proposed so far offer guarantees only in the smooth ( C^{1, 1}) setting. Indeed, the lfs of Svanishes at points where Sis not smooth. In , we introduce a new measurable quantity, called the Lipschitz radius, which plays a role similar to that of lfs in the smooth setting, but turns out to be well-defined and positive on a large class of (smooth and non smooth) shapes. Given a surface S, the k-Lipschitz radius at a point p, noted lr _k( p), is the radius of the largest ball Bcentered at psuch that SBis the graph of a k-Lipschitz bivariate function. The class of surfaces with positive k-Lipschitz radius includes piecewise smooth surfaces provided that the normal variation around singular points is not too large.

Our main result is that, if Shas a positive k-Lipschitz radius (for some small enough k) and Eis a sample of Ssuch that any point pS is at distance less than a fraction of lr _k( p), then we obtain similar guarantees as in the smooth setting. More precisely, we show that the Delaunay triangulation of Erestricted to Sis a 2-manifold isotopic to Slying at Hausdorff distance O( )from S, provided that its facets are not too skinny.

We further extend our results to loose -samples. Loose -sample have been introduced to analyze smooth surface meshing algorithms . This notion is weaker than the notion of -sample. Nevertheless, we show that the previous results still hold if the sample Eis only a loose -sample. As a straightforward application, the Delaunay refinement algorithm we proved correct for smooth surfaces works fine and offers the same topological and geometric guarantees for Lipschitz surfaces. Specifically, the output of the algorithm is a PL surface with an optimal number of vertices that is isotopic to S, and lies at Hausdorff distance O( )from  S.

To the best of our knowledge, this is the first provably correct algorithm for meshing nonsmooth surfaces.

Conformal -shapes

We define a new filtration of the Delaunay triangulation of a finite set of points in , similar to the alpha shape filtration. The new filtration is parameterized by a local scale parameter instead of the global scale parameter in alpha shapes. Since our approach shares many properties with the alpha shape filtration and the local scale parameter conforms to the local geometry we call it conformal alpha shape filtration. The local scale parameter is motivated from applications and previous algorithms in surface reconstruction. We show how conformal alpha shapes can be used for surface reconstruction of non-uniformly sampled surfaces, which is not possible with alpha shapes.

Isosurfaces of 3D images are usually extracted using the marching cube algorithm. For high resolution images, the resulting surfaces can contain a very large number of triangles, which makes their processing difficult and slow. In this work  , we combine the marching cube algorithm for surface extraction and the edge contraction algorithm for surface simplification in lock step, which avoids the costly step of storing the entire triangulation. Beyond this basic strategy, we introduce refinements to prevent artifacts in the resulting triangulation, first, by carefully monitoring the amount of simplification during the process and, second, by driving the simplification toward a compromise between shape approximation and mesh quality. We have implemented the algorithm and used extensive computational experiments to document the effects of various design options and to further fine-tune the algorithm.

F. Labelle and J. Shewchuk have proposed a discrete definition of anisotropic Voronoi diagrams. These diagrams are parametrized by a metric field. Under mild hypotheses on the metric field, such Voronoi diagrams can be refined so that their dual is a triangulation, with elements shaped according to the specified anisotropic metric field.

We propose an alternative view of the construction of these diagrams and a variant of Labelle and Shewchuk's meshing algorithm . This variant computes the Voronoi vertices using a higher dimensional power diagram and refines the diagram as long as dual triangles overlap. We see this variant as a first step toward a 3-dimensional anisotropic meshing algorithm.

We introduce a novel Delaunay-based variational approach to isotropic tetrahedral meshing  . To achieve both robustness and efficiency, we minimize a simple mesh-dependent energy through global updates of both vertex positions and connectivity. Building upon the function approximation approach, we optimize the mesh so as to minimize the L1 distance between an isotropic quadratic function and its linear interpolation on the mesh. Mesh design is controlled through a gradation smoothness parameter and selection of the desired number of vertices. We provide the foundations of our approach by explaining both the underlying variational principle and its geometric interpretation. We demonstrate the quality of the resulting meshes (see Figure  ).

This work introduces a three-dimensional mesh generation algorithm for domains bounded by smooth surfaces , . The algorithm combines a Delaunay-based surface mesher  with a Ruppert-like volume mesher, to get a greedy algorithm that samples the interior and the boundary of the domain at once. The algorithm constructs provably-good meshes, it gives control on the size of the mesh elements through a user-defined sizing field, and it guarantees the accuracy of the approximation of the domain boundary. A noticeable feature is that the domain boundary has to be known only through an oracle that can tell whether a given point lies inside the object and whether a given line segment intersects the boundary. This makes the algorithm generic enough to be applied to a wide variety of objects, ranging from domains defined by implicit surfaces to domains defined by level-sets in 3D grey-scaled images or by point-set surfaces (see Fig.  ).

Given a smooth surface, a blue (red) ridge is a curve such that at each of its points, the maximum (minimum) principal curvature has an extremum along its curvature line. Ridges are curves of extremalcurvature and therefore encode important informations used for segmentation, registration, matching and surface analysis. State of the art methods for ridge extraction either report red and blue ridges simultaneously or separately —in which case a local orientation procedure of principal directions is needed, but no method developed so far certifies the topology of the curves reported.

On the way to developing certified algorithms independent from local orientation procedures, we make the following fundamental contribution . For any smooth parametric surface, we exhibit the implicit equation P= 0of the singular curve Pencoding all ridges of the surface (blue and red), and show how to recover the colors from factors of . Exploiting P= 0, we also derive a zero dimensional system coding the so-called turning points, from which elliptic and hyperbolic ridge sections of the two colors can be derived. (Elliptic sections correspond to a maximum (minimum) of the maximum (minimum) principal curvature, while hyperbolic sections correspond to a minimum (maximum) of the maximum (minimum) principal curvature.) Both contributions exploit properties of the Weingarten map of the surface and require computer algebra. Algorithms exploiting the structure of for algebraic surfaces are developed in a companion paper .

Ridges of a smooth surface are curves of extremalcurvature and encode important informations used in surface analysis or segmentation. But reporting them requires manipulating third and fourth order derivatives —whence numerical difficulties. Additionally, ridges have self-intersections and complex interactions with the umbilics of the surface —whence topological difficulties.

In this context, we make two contributions for the computation of ridges of polynomial parametric surfaces . First, by instantiating to the polynomial setting a global structure theorem of ridge curves proved in a companion paper , we develop the first algorithm to produce an approximation with certified topology of the curve encoding all the ridges of the surface. The algorithm exploits the singular structure of umbilics and intersection points between red and blue ridges, and reduces the problem to solving zero dimensional systems using Gröbner basis. Second, for cases where the zero-dimensional systems cannot be practically solved, we develop a certified plot algorithm at any fixed resolution —a pixel being lit iff its closure is intersected by the curve .

These contributions are respectively illustrated for Bézier surfaces of degree four and five. See Figs. . and .

The idea of topological persistence is to study the evolution of the topology of the sublevel-sets of a real function defined on a topological space as the threshold increases. It turns out that this information can be encoded as a certain set of intervals called persistence intervals. Each interval represents the life-time of a topological feature —for instance a handle or a connected component— in the evolution of sublevel-sets. Associating with each persistence interval a point in the plane whose coordinates are the bounds of the interval, we obtain the persistence diagramof the function. In this work, we prove that the persistence diagrams is stable when the function is corrupted by noise . We apply this result to the comparison and classification of shapes, and to the estimation of the homology of a subset of a metric space from sample points.

In this paper  , we bound the difference between the total mean curvatures of two closed surfaces in in terms of their total absolute curvatures and the Fréchet distance between the volumes they enclose. Unlike previous results, and maybe surprisingly, the closeness between the normals to the two surfaces is not explicitly required. The proof combines the stability property for persistence diagrams  with integral-geometric methods. We also bound the difference between the lengths of two curves using the same methods, which gives a generalization of Fàry's theorem.

Our work on the design of compact structures for representing a triangulation in main memory has been extended to allow dynamic updates of the triangulation . The main idea remains a hierarchical structure, splitting a triangulation of ntriangles in O(log ²n)small pieces, each of them being split again in O(log n)tiny pieces of size <log n. The difference mainly involves the use of various dynamic arrays to store connectivity information between these pieces and new levels of indirection to be able to manage the variable memory size used to store these pieces. The result is that a triangulation can be stored with a cost of 2.17× nbits plus a sublinear term and allowing dynamic update of the triangulation in O(log ²n)per insertion, flip or deletion of a degree 3 vertex , .

In the spirit of the mostly theoretical work above, we investigate more practical solution that could be used in cgal. Classical storage of triangulation usually involve 12 pointers per triangle that is 12×32× nbits. From a theoretical point of view, going to 12×log n× nbits is trivial, but in practice it is much more difficult because reading and writing memory by words of variable size is not really what is expected by the operating system. We have implemented and experimented a new triangulation data structure in cgalwhich allows this kind of operation. We save about 50% of the memory (depending on the triangulation size) with a big increase of the CPU time.

This paper provides a detailed experimental study of an interface model developed in the companion article F. Cazals and F. Proust, Revisiting the description of Protein-Protein interfaces. Part I: algorithms. Our experimental study is concerned with the usual database of protein-protein complexes, split into five families (Proteases, Immune system, Enzyme Complexes, Signal transduction, Misc.) Our findings, which bear some contradictions with usual statements are the following: (i) Connectivity properties incur important variations across families. These properties are sensitive to water molecules and their variations upon consideration of structural water quantify the filling of packing defects by water molecules. (ii) The model of interfaces as a hydrophilic rim and a hydrophobic core is not general. (iii) At the interface scale, curvature properties correlate with the interface surface area, while locally, the absolute mean curvature follows a bimodal distribution. (iv) About 10% of interfaces consists of several connected components, and this multi-patch structure is independent from structural water. Almost all interfaces feature holes of significant size filled by structural water. Stable crystallographic water molecules play a prominent in reconnecting disconnected interfaces. (v) The chemical composition of interfaces in terms or pairwise contacts features a constant ratio of undetermined interactions, with subtle inter-families variations of determined interactions.

Overall, these conclusions shed some light on which structural parameters are most relevant to describe protein-protein interactions.

We present a new method to compute interactive reflections on curved objects. The approach creates virtual reflected objects which are blended into the scene. We use a property of the reflection geometry which allows us to efficiently and accurately find the point of reflection for every reflected vertex, using only reflector geometry and normal information. This reflector information is stored in a pair of appropriate cube-maps, thus making it available during rendering. The implementation presented achieves interactive rates on reasonably sized scenes. In addition,we introduce an interpolation method to control the accuracy of our solution depending on the required frame rate.

We propose an algorithm for placement of streamlines from two-dimensional steady vector or direction fields  . Our method consists of placing one streamline at a time by numerical integration starting at the furthest away from all previously placed streamlines. Such a farthest point seeding strategy leads to high quality placements by favoring long streamlines, while retaining uniformity with the increasing density. Our greedy approach generates placements of comparable quality with respect to the optimization approach from Turk and Banks, while being 200 times faster. Simplicity, robustness as well as efficiency is achieved through the use of a Delaunay triangulation to model the streamlines, address proximity queries and determine the biggest voids by exploiting the empty circle property. Our method handles variable density and extends to multiresolution.

The goal of this study is to optimize pupil configurations for extended source imaging based on optical interferometry.

The underlying problem can be formally stated as follows: given an objective Osupposed to be a disk, how to place a system of circular pupils such that its auto-correlation support covers entirely the objective where and ?

In , we show that power diagrams can help solving special cases of the general optimization problem. In particular, when the centers of the pupils are given, we provide efficient algorithms to minimize the total area of the pupils under the constraint that covers the objective. Heuristics are also provided for the general problem where the positions of the pupils are unknown.

We have submitted a proposal of specification of interval arithmetic to the C++ standardization committee  . The goal of this submission are several : have compiler writers aware of the wide use of interval arithmetic (by numerous INRIA projects including GEOMETRICA) and so we hope that they will provide specific optimizations for it, and have a standard interface for interval computations. Some industry members of the C++ standardization committee like Sun are also very interested in getting interval arithmetic standardized.

Our proposal is a basic template class parameterized by a floating point type, and provide some functions around it. Due to implementation cost constraints, we had to make choices between the simplicity of the implementation, and functionality. This appears for example in the choice we made for the specification of comparison operators between intervals. We are working on a revision of our proposal for the next standardization meeting in 2006, integrating the comments we got from the committee members.

We have started implementing a scilabtoolbox dedicated to geometric computing, based on cgal. A first beta version of this toolbox, to be released in 2005, provides the basic Delaunay triangulations in any dimension, as well as various 2D meshing algorithms.