Geometric computing plays a central role in most engineering activities: geometric modelling, computer aided design and manufacturing, computer graphics and virtual reality, scientific visualization, geographic information systems, molecular biology, fluid mechanics, and robotics are just a few well-known examples. The rapid advances in visualization systems, networking facilities and 3D sensing and imaging make geometric computing both dominant and more demanding concerning effective algorithmic solutions.

Computational geometry emerged as a discipline in the seventies and has met with considerable success in resolving the asymptotic complexity of basic geometric problems including data structures, convex hulls, triangulations, Voronoi diagrams, geometric arrangements and geometric optimisation. However, in the mid-nineties, it was recognized that the applicability in practice of the computational geometry techniques was far from satisfactory and a vigorous effort has been undertaken to make computational geometry more effective. The prismeproject together with several partners in Europe took a prominent role in this research and in the development of a large library of computational geometry algorithms, cgal.

geometricaaims at pursuing further the effort in this direction and at building upon the initial success. Its focus is on effective computational geometry with special emphasis on
*curves and surfaces*. This is a challenging research area with a huge number of potential applications in almost all application domains involving geometric computing.

The overall objective of the project is to give effective computational geometry for curves and surfaces solid mathematical and algorithmic foundations, to provide solutions to key problems and to validate our theoretical advances through extensive experimental research and the development of software packages that could serve as steps towards a standard for safe and effective geometric computing.

The research conducted by geometricafocuses on three main directions:

- fundamental geometric data structures and algorithms

- robust computation and advanced programming,

- shape approximation and mesh generation.

geometricais pursuing long standing research on fundamental geometric data structures and algorithms. geometricahas a large expertise in Voronoi diagrams and Delaunay triangulations, randomized algorithms, combinatorial geometry and related fields. Recently, we devoted efforts to developing the field of computational geometry beyond linear objects. We are especially interested in developing a theory of curved Voronoi diagrams. Such diagrams allow to model growing processes and have important applications in biology, ecology, chemistry and other fields. They also play a role in some optimization problems and in anisotropic meshing. Euclidean Voronoi diagrams of non punctual objects are also non affine diagrams. They are of particular interest in robotics, CAD and molecular biology. Even for the simplest diagrams, e.g. Euclidean Voronoi diagrams of lines, triangles or spheres in 3-space, obtaining tight combinatorial bounds and efficient algorithms are difficult research questions. In addition, effective implementations require to face specific algebraic and arithmetic questions. Working out carefully the robustness issues is a central objective of geometrica(see below).

In past years, the main objective of computational geometry has been the design of time efficient algorithms, either from the theoretical point of view of the asymptotic complexity or the more practical of running efficient benchmarks. Surprisingly less interest has been devoted to improve the space behavior of such algorithms although the problem may become of importance when the main memory is not enough and the system has to swap to find the extra memory; this may happen either for massive data or for small memory devices such as PDA. geometricaintends to attack this problem from several aspects:

— the compression of geometrical objects for storage or network transmission purposes

— the design of algorithms accessing locally the memory to reduce (but not remove) the swapping in memory

— the design of new data-structures to represent geometrical objects using less memory.

For all these aspects we are interested in both the theoretical asymptotic sizes involved and the practical aspect for reasonable input size. Such a distinction between asymptotic behavior and practical interest may appear strange since we are interested in massive data, but even for a big mesh of several millions points which can be called ``massive data'' we are still far from the asymptotic behavior of some theoretical structures.

An implementation of a geometric algorithm is called
*robust*if it produces a valid output for all inputs. Geometric programs are notorious for their non-robustness due to two reasons: (1) Geometric algorithms are designed for a model of computation where real numbers are dealt with exactly and (2) geometric algorithms are frequently only
formulated for inputs in general position. As a result, implementations may crash or produce nonsensical output. This is observed in all commercial
cad-systems.

The importance of robustness in geometric computations has been recognized for a long time, but significant progress was made only in recent years.
geometricaheld a central role in this process, including advances regarding the
*exact computation paradigm*. In this paradigm, robustness is achieved by a combination of three methods:
*exact arithmetic*,
*dedicated arithmetic*and
*controlled rounding*.

In addition to pursuing research on robust geometric computation, geometricais an active member of a European consortium that develops a large library named cgal. This library makes extensive use of generic programming techniques and is both a unique tool to perform experimental research in Computational Geometry and a comprehensive library for Geometric Computing. The startup company GeometryFactory has been started in January 2003 to commercialize components from cgaland to offer services for geometric applications.

Complex shapes are ubiquitous in robotics (configuration spaces), computer graphics (animation models) or physical simulations (fluid models, molecular systems). In all these cases, no natural
*shape space*is available or when such spaces exist they are not easily dealt with. When it comes to performing calculations, the objects under study must be discretized. On the other hand, several application areas such as Computer Aided Geometric Design or medical imaging require
reconstructing
3
Dor
4
Dshapes from samples.

The questions afore-mentioned fall in the realm of
*geometric approximation theory*, a topic
geometricais actively involved in. More precisely, the generation of samples, the definition of differential quantities (e.g. curvatures) in a discrete setting, the geometric and topological control of approximations, as well as multi-scale
representations are investigated. Connected topics of interest are also the progressive transmission of models over networks and their compression.

Surface mesh generation and surface reconstruction have received a great deal of attention by researchers in various areas ranging from computer graphics through numerical analysis to computational geometry. However, work in these areas has been mostly heuristic and the first theoretical
foundations have been established only recently. Quality mesh generation amounts to finding a partition of a domain into linear elements (mostly triangles or quadrilaterals) with topological and geometric properties. Typically, one aims at constructing a piecewise linear (PL) approximation
with the ``same'' topology as the original surface (same topology may have several meanings). In some contexts, one wants to simplify the topology in a controlled way. Regarding the geometric distance between the surface and its PL approximation, different measures must be considered:
Hausdorff distance, errors on normals, curvatures, areas etc. In addition, the shape, angles or size of the elements must match certain criteria. We call
*remeshing*the techniques involved when the input domain to be discretized is itself discrete. The input mesh is often highly irregular and non-uniform, since it typically comes as the output of a surface reconstruction algorithm applied to a point cloud obtained from a scanning device.
Many geometry processing algorithms (e.g. smoothing, compression) benefit from remeshing, combined with uniform or curvature-adapted sampling.
geometricaintends to contribute to all aspects of this matter, both in theory and in practice.

Volumetric mesh generation consists in triangulating a given three-dimensional domain with tetrahedra having a prescribed size and shape. For instance, the tetrahedra in a general purpose mesh should be as regular as possible (isotropic case), whereas they should be elongated in certain
directions for problem specific meshes. Volumetric mesh generation often makes use of surface mesh generation techniques (
*e.g.*to approximate the boundary of the domain or interfaces between two media). Thanks to its strong experience with Delaunay triangulations,
geometricarecently made several contributions to the generation of volumetric meshes, and intends to pursue in this direction.

Modeling 3D shapes is required for all visualization applications where interactivity is key since the observer can change the viewpoint and get an immediate feedback. This interactivity enhances the descriptive power of the medium significantly. For example, visualization of complex molecules helps drug designers to understand their structure. Multimedia applications also involve interactive visualization and include e-commerce (companies can present their product realistically), 3D games, animation and special effects in motion pictures. The uses of geometric modeling also cover the spectrum of engineering, computer-aided design and manufacture applications (CAD/CAM). More and more stages of the industrial development and production pipeline are now performed by simulation, due to the increased performance of numerical simulation packages. Geometric modeling therefore plays an increasingly important role in this area. Another emerging application of geometric modeling with high impact is medical visualization and simulation.

In a broad sense, shape reconstruction consists of creating digital models of real objects from points. Example application areas where such a process is involved are Computer Aided Geometric Design (making a car model from a clay mockup), medical imaging (reconstructing an organ from medical data), geology (modeling underground strata from seismic data), or cultural heritage projects (making models of ancient and or fragile models or places). The availability of precise and fast scanning devices has also made the reproduction of real objects more effective such that additional fields of applications are coming into reach. The members of geometricahave a long experience in shape reconstruction and contributed several original methods based upon the Delaunay and Voronoi diagrams.

Two of the most prominent challenges of the post-genomic era are to understand the molecular machinery of the cell and to develop new drug design strategies. These key challenges require the determination, understanding and exploitation of the three-dimensional structure of several classes of molecules (nucleic acids, proteins, drugs), as well as the elucidation of the interaction mechanisms between these molecules.

These challenges clearly involve aspects from biology, chemistry, physics, mathematics and computer science. For this latter discipline, while the historical focus has been on text and pattern matching related algorithms, the amount of structural data now available calls for geometric methods. At a macro-scopic scale, the classification of protein shapes, as well as the analysis of molecular complexes requires shape description and matching algorithms. At a finer scale, molecular dynamics and force fields require efficient data-structures to represent solvent models, as well as reliable meshes so as to solve the Poisson-Boltzmann equation.

Meshes are the basic tools for scientific computation. Unstructured meshes allow to mesh complex shapes and to refine locally the mesh, which may be required because of the geometry or in order to increase the precision of the computation. geometricacontributes to 2D and 3D meshes, and also to surface meshes. The methods are mostly based on Delaunay triangulations, Voronoi diagrams and their variants. Affine diagrams are well-suited for volume element methods. Non affine diagrams are especially important in the context of anisotropic meshes. Anisotropic quadrilateral meshes are also of interest.

The emerging demand for visualizing and simulating 3D geometric data in networked environments has motivated research on representations for such data. Slow networks require data compression to reduce the latency, and progressive representations to transform 3D objects into streams
manageable by the networks. The members of
geometricahave contributed several original compression methods for surface and volume meshes
. We investigate both single-rate and progressive compression depending on whether the model is intended to be
decoded during, or only after, the transmission. Progressive compression is closely related to both approximation and information theory, aimed at trading off data size for approximation accuracy (so-called rate-distortion tradeoff). We cast this problem into that of
*shape compression*.

cgalis a C++ library of geometric algorithms developed initially within two European projects (project ESPRIT IV LTR CGAL december 97 - june 98, project ESPRIT IV LTR GALIA november 99 - august 00) by a consortium of eight research teams from the following institutes: Universiteit Utrecht, Max-Planck Institut Saarbrücken, INRIA Sophia Antipolis, ETH Zürich, Tel Aviv University, Freie Universität Berlin, Universität Halle, RISC Linz. The goal of cgalis to make the solutions offered by the computational geometry community available to the industrial world and applied domains.

The
cgallibrary consists in a kernel, a basic library and a support library. The kernel is made of classes that represent elementary geometric objects (points, vectors, lines, segments, planes, simplices, isothetic boxes...), as well as affine
transformations and a number of predicates and geometric constructions over these objects. These classes exist in dimensions 2 and 3 (static dimension) and
d(dynamic dimension). Using the template mechanism, each class can be instantiated following several representation modes : we can choose between Cartesian or homogeneous coordinates, use different types to store the coordinates, and use reference counting or not. The
kernel also provides some robustness features using some specifically-devised arithmetic (interval arithmetic, multi-precision arithmetic, static filters...).

The basic library provides a number of geometric data structures as well as algorithms. The data structures are polygons, polyhedra, triangulations, planar maps, arrangements and various search structures (segment trees,
d-dimensional trees...). Algorithms are provided to compute convex hulls, Voronoi diagrams, boolean operations on polygons, solve certain optimization problems (linear, quadratic, generalized of linear type). Through class and function templates, these algorithms can be
used either with the kernel objects or with user-defined geometric classes provided they match a documented interface (concept).

Finally, the support library provides random generators, and interfacing code with other libraries, tools, or file formats (Ascii files, QT or LEDA Windows, OpenGL, Open Inventor, Postscript, Geomview, scilab...).

geometricais particularly involved in the arithmetic issues that arise in the treatment of robustness issues, the kernel, triangulation packages and their close applications such as alpha shapes, general maintainance...

cgalis about 500,000 lines of code and supports various platforms: GCC (Linux, Mac OS, Solaris, Irix, Cygwin...), MipsPro (IRIX), SunPro (Solaris), Visual C++ (Windows), Intel C++... Version 3.1 has been released on december 21st, 2004. The previous release, cgal3.0, has been downloaded 14000 times from our web site, during the 11 months period where it was the main version.

In collaboration with Frank Da and Andreas Fabri. http://cgal.inria.fr/Reconstruction/.

The surface reconstruction algorithm developed by D. Cohen-Steiner and F. Da using
cgalis available as a
*web service*. Via the web, the user uploads the point cloud data set to the server and obtains a
Vrmlfile of the reconstructed surface, which gets visualized in the browser of the user. This allows the user to check if the algorithm fits his/her needs, to learn how to adjust the parameters, before contacting
Inriato obtain an executable. It also provides us with the opportunity to collect real-world data sets used for testing and improving our reconstruction algorithms.

The presentation of our new results follows the three main directions recalled in Section

– fundamental geometric data structures and algorithms

– robust computation and advanced programming,

– shape approximation and mesh generation.

Contributions on the first item are the construction of Voronoi diagrams of spheres, several results related to visibility and graph algorithms for reporting maximal cliques.

Work on the second research direction includes the design and a preliminary implementation of a certified curved kernel for cgal. In addition, to further extend the robustness and the efficiency of cgal, geometric constructions are now filtered in a way similar to predicates.

Work on the third axis gained momentum and is subdivided in Surface Approximation, Mesh generation, Computation and Stability of Geometric Features, Shape Coding and Compression.

Lastly, we report on some applications in Molecular Biology, Visualization and Signal Processing.

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.

In collaboration with H. Brönnimann (Polytechnic Univ. Brooklyn), V. Dujmović and S. Whitesides (Univ. Mac Gill), H. Everett, M. Glisse, X. Goaoc and S. Lazard (INRIA-Vegas), S. Hornus (INRIA-Artis), H.-S. Na (Soongsil Univ., Seoul), F. Sottile (Texas A&M Univ.) and S. Wismath (Univ. Lethbridge).

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
*free*if it intersects no other objects, it is called
*free line segment*if it intersects no objects between the points where the line touch the four objects and
*occluded*otherwise.

First, we prove that the lines tangent to four (possibly intersecting) convex polyhedra with
nedges in total, form
(
n^{2})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,
(
n^{2}k^{2})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.

In collaboration with H. Everett, S. Lazard, M. Pentcheva (INRIA-Vegas) and S. Wismath (U. Lethbridge).

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 intersecting**polylines 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_{n}inside a box of volume
O(
n^{2.5}). The lower bound for non intersecting 3D drawing of
K_{n}is known to be
(
n^{2}), 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}
,
.

In collaboration with Chinmay Karande, IIT Bombay (3rd year student).

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
*pathological*worst-cases.

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.

In collaboration with Chinmay Karande, IIT Bombay (3rd year student).

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)
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.* c-connected cliques*in product graphs, a problem for which an algorithm has been presented in a recent paper

In collaboration with Monique Teillaud, Constantinos Tsirogiannis and Ilya Suslov (latter two are student interns).

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.

In collaboration with Andreas Fabri (GeometryFactory).

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.

In collaboration with Leo Guibas (Stanford University).

We consider the problem of discovering a
C^{2}unknown 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.

In collaboration with J. Giesen (ETH Zurich), M. Pauly (ETH Zurich), A. Zomorodian (Stanford University).

In collaboration with Dominique Attali (LIS, Grenoble) and Herbert Edelsbrunner (Duke University).

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.

In collaboration with M.Desbrun (California Institute of Technology).

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. ).

In collaboration with F. Rouillier (INRIA-SALSA, Rocquencourt) and J-C. Faugère (INRIA-SALSA, Rocquencourt).

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
*extremal*curvature 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
.

In collaboration with F. Rouillier (INRIA-SALSA) and J-C. Faugère (INRIA-SALSA Rocquencourt).

Ridges of a smooth surface are curves of
*extremal*curvature 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 .

In collaboration with Herbert Edelsbrunner and John Harer (Duke University).

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 diagram*of 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 collaboration with Herbert Edelsbrunner (Duke University).

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.

In collaboration with Gilles Schaeffer (Laboratoire d'Informatique de l'École Polytechnique, LIX)

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
^{2}n)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
^{2}n)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.

In collaboration with F. Proust and J. Janin (CNRS, Orsay).

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

In collaboration with Pau Estella, Ignacio Martin (U. Girona) George Drettakis (INRIA-REVES) and Dani Tost (UPC Barcelona)

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 collaboration with H. Brönnimann (Polytechnic University Brooklyn), and G. Melquiond (ARENAIRE).

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.

**Surface Mesher**(participants: Laurent Rineau, Steve Oudot, Mariette Yvinec, in collaboration with David Rey (from DREAM staff) and Andreas Fabri (from GeometryFactory)). This new
cgalpackage provides an implementation of the algorithm developped by Boissonnat and Oudot
to mesh smooth surfaces. The current version of the package handles implicit surfaces and level-sets in 3D
grey-scaled images. The interface offers a few parameters to achieve a fine tuning according to the user needs with respect to the quality of the mesh elements and the accuracy of the surface approximation. The output mesh can also conform to a user defined sizing field. The code
follows a C++ generic software design for Delaunay refinement processes proposed by Laurent Rineau.

**Planar Parameterization of Triangulated Surfaces**(participants: Laurent Saboret and Pierre Alliez, in collaboration with B. Lévy (INRIA-ALICE project)). We are elaborating upon a
cgalpackage for planar parameterization of triangulated surface meshes. The package implements some of the state-of-the-art parameterization methods, such as LSCM (Lévy et al. 02), fixed or free boundary conformal maps (Eck et al. 95, Meyer et
al. 02), Authalic (Meyer et al. 02), mean value coordinates (Floater 03) or Tutte barycentric mapping with uniform weights (Tutte 63). It supports two mesh
cgaldata structures and three sparse linear solvers (OpenNL, TAUCS, SuperLU). A first submission of the package to the
cgaleditorial board is planned for end of 2005.

**Placement of Streamlines**(participant: Abdelkrim Mebarki). We are elaborating upon a
cgalpackage for placement of streamlines in 2D which implements the algorithm presented at Visualization 05
. The input flow field is given as a vector or direction field defined onto a closed planar domain. The
algorithm saturates the domain with a set of streamlines which are everywhere tangential to the input flow field. The streamline density is in accordance with either a user-defined input parameter or with a density field. The package comes with an interactive demo available on both
Linux and Windows platforms.

The initial development phase of the cgallibrary has been made by a European consortium. In order to achieve the transfer and diffusion of cgalin the industry, a company called GeometryFactory has been founded in January 2003 by Andreas Fabri ( http://www.GeometryFactory.com).

The goal of this company is to pursue the development of the library and to offer services in connection with cgal(maintenance, support, teaching, advice). GeometryFactory is a link between the researchers from the computational geometry community and the industrial users.

It offers licenses to interested companies, and provides support. There are contracts in various domains such as CAD/CAM, medical applications, GIS, computer vision...

GeometryFactory is keeping close contacts with the original consortium members, and in particular with geometrica.

In collaboration with P. Blanc and F. Falzon (Alcatel Alenia Space).

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

The motivation for this work comes from the observation of the Earth from a geostationary orbit (i.e. at a distance of km) with a resolution of 1 m. A simple calculus shows us that we would need a telescope having a diameter of approximately 20 m for an optical wavelength of nm. Needless to say such an instrument dimension is not adapted to the observation from space and the use of interferometric telescopes (Optical Aperture Synthesis, OAS) is to be considered in this case.

Initial results based on Laguerre diagrams have proven useful in this context . They will be further extended as part of T. Nguyen's PhD thesis.

We are a member of Geocompan initiative from the national «Action Concertée Incitative Masses de Données». The project involves

— the «Laboratoire d'Informatique de l'Ecole Polytechnique (LIX), CNRS, Ecole Polytechnique»,

— the «Laboratoire Bordelais de Recherche en Informatique (LaBRI), CNRS, Université Bordeaux I»,

— the «Service de Physique Théorique, CEA Saclay» and the geometricateam.

The project investigates compression schemes and compact data structures devoted specifically to geometrical objects.

Project web page: http://www.lix.polytechnique.fr/Labo/Gilles.Schaeffer/GeoComp/

Also involved: Monique Teillaud.

We have started an INRIA associated team with Chee Yap (New York University) and Hervé Brönnimann (Polytechnic University Brooklyn), around the subjects of generic programming and robustness of geometric algorithms. This work includes the specification of algorithms in terms of concepts of geometries. It also includes the interface between algorithms and data structures, as well as collaborations on robustness issues around curved objects.

inria(teams galaadand geometrica) participates to the IST project ACS.

– Acronym : ACS, numéro IST-006413

– Title : Algorithms for complex shapes with certified topology and numerics.

– Specific program : IST

– RTD (FET Open)

– Starting date : may 1st, 2005 - Duration : 3 years

– Participation mode of Inria : Participant

– Other participants : Rijksuniversiteit Groningen, Eidgenössische Technische Hochschule Zürich, Freie Universität Berlin, Institut National de Recherche en Informatique et Automatique, Max-Planck-Gesellschaft zur Förderung der Wissenschaften e.V., National Kapodistrian University of Athens, Tel Aviv University, GeometryFactory Sarl

The ACS project aims at advancing the state of the art in computing with complex shapes. Current technolgy can cope well with curves in the plane and smooth surfaces in three-dimensional space. We want to address a larger class of shapes, including piecewise smooth surfaces, surfaces with singularities, as well as manifolds of codimension larger than one in moderately high dimension.

Increasingly demanding applications require efficient and robust algorithms for complex shapes. Topics that arise and that we address are shape approximation (including meshing and simplification), shape learning (including reconstruction and feature extraction), as well as robust modeling (including boolean operations). Our work on these topics will be closely intertwined with basic research on shape representations.

A unique and ambitious feature of our approach is the guaranteed quality of all data structures and algorithms we plan to develop. Through certified topology and numerics, we will be able to prove that the output is topologically and numerically consistent, according to prespecified criteria. A software prototype, dealing with a restricted class of complex shapes, will demonstrate the feasibility of our techniques in practice.

The web site of the project includes a detailled description of the objectives and all results http://acs.cs.rug.nl.

inriais part of the Network of Excellence:

– Acronym : AIM@SHAPE

– Title: Advanced and Innovative Models And Tools for the development of Semantic-based systems for Handling, Acquiring, and Processing Knowledge Embedded in multidimensional digital objects).

– Reference: 506766

– Start Date: 2004-01-01

– Duration: 48 months

– Contract Type: Network of Excellence

– Action Line: Semantic-based knowledge systems

– Project Funding: 5.74 Million Euros.

– Other participants :

– IMATI, Genova, Italy.

– University of Genova, Italy.

– EPFL, Lausanne, Switzerland.

– Fraunhofer Institute, Germany.

– INPG, France.

– Center for Research and Technology, Greece.

– University of Geneva, Switzerland.

– SINTEF, Norway.

– TECHNION, Israel.

– Weizmann Institute, Israel.

– Utrecht University, Netherlands.

The mission of AIM@SHAPE is to advance research in the direction of semantic-based shape representations and semantic-oriented tools to acquire, build, transmit, and process shapes with their associated knowledge. We foresee a generation of shapes in which knowledge is explicitly represented and, therefore, can be retrieved, processed, shared, and exploited to construct new knowledge. The attainment of a new vision of shape knowledge is achieved by: the formalisation of shape knowledge and the definition of shape ontologies in specific contexts; the definition of shape behaviours which formalise the interoperability between shapes; the delineation of methods for knowledge-based design of shapes and the definition of tools for semantics-dependent mapping of shapes. The web site of the network includes a detailled description of the objectives and some results http://www.aim-at-shape.net.

- J-D. Boissonnat is a member of the editorial board of
*Theoretical Computer Science*,
*Algorithmica*,
*International Journal of Computational Geometry and Applications*,
*Computational Geometry : Theory and Applications*, and
*The Visual Computer*.

- M. Yvinec is a member of the editorial board of
*Journal of Discrete Algorithms*.

- S. Pion and M. Yvinec are members of the cgaleditorial board.

- Jean-Daniel Boissonnat was a member of the program committees of Solid Modelling International 2005, of the ACM Symposium on Solid and Physical Modeling 2005 and of the Eurographics Symposium on Geometry Processing 2005.

- Jean-Daniel Boissonnat was a member of the program committee of the Journée Recherche Industrie «Des données et modèles 3D aux calculs et à l'exploitation de résultats», INRIA Rocquencourt, février 2005.

- Pierre Alliez was member of the paper committee of the third Eurographics Symposium on Geometry Processing 2005, Eurographics 2005 and Pacific Graphics 2005. He was also member of the Video and Multimedia presentation program committee for the 21st Annual ACM Symposium on Computational Geometry (SOCG) 2005.

- Frédéric Cazals was member of the paper committee of the Eurographics Symposium on Geometry Processing 2005, of the Symposium on Point Based Graphics 2005, and of the ACM Symposium on Solid and Physical Modeling 2005.

- Pierre Alliez was a referee of the Ph.D. thesis committee of Raphaèle Balter (Université de Rennes) and Christian Rössl (MPII Saarbrucken). He was a member of the thesis committee of Marie-Claude Frasson (Université de Nice Sophia-Antipolis), Gabriel Peyré (Ecole Polytechnique) and Guillaume Lavoué (Université de Lyon).

- Jean-Daniel Boissonnat was a referee of the Habilitation dissertations of F. Chazal (Université de Bourgogne) and J. Giesen (ETH Zurich). He was a referee of the Ph.D. thesis committees of M. Flandrin (IFP) and K. Nouioua (Université de Méditerranée).

- Olivier Devillers was a referee of the Ph.D. thesis committee of Geoffroy Lauvaux (Université de Reims Champagne Ardennes) and Ali Asghar Khanban (Imperial college, London). He was a member of the thesis committee of Thomas Lewiner (Paris VI) and referee of the HDR committee of Gilles Schaeffer (Université de Bordeaux).

- David Cohen-Steiner was a referee of the Ph.D. thesis committee of Dmitriy Morozov (Duke University).

-Jean-Daniel Boissonnat was chairman of the Projects Committee of INRIA Sophia-Antipolis till august 2005. He is chairman of the Evaluation Board of INRIA since november 2005.

- Frédéric Cazals is member of the INRIA Scientific Steering Committee of INRIA (COST).

- Mariette Yvinec is member of the « Comité des utilisateurs des moyens informatiques de recherche de l'INRIA Sophia-Antipolis » (CUMIR)

- Agnès Bessière is member of the « Comité des utilisateurs des moyens informatiques des services de l'INRIA Sophia-Antipolis » (CUMIS)

- Olivier Devillers is member of the committee for « détachements » at INRIA Sophia-Antipolis.

- Agnès Bessière, Olivier Devillers and Pierre Alliez are members of the « comité de centre » at INRIA Sophia-Antipolis.

- Sylvain Pion is a member of the « Commission du Développement Logiciel » (CDL) at INRIA Sophia-Antipolis.

- Pierre Alliez is member of the « Comité des cours et colloques » at INRIA Sophia-Antipolis.

- Jean-Daniel Boissonnat is member of the « Commission de spécialistes » of the Ecole Normale Supérieure de Paris.

- Frédéric Cazals is member of the « Commission de spécialistes » of the Mathematics Department of the University of Bourgogne, Dijon, France.

- Olivier Devillers is member of the «Conseil scientifique de l'école doctorale STIC» at Nice University.

- Sylvain Pion is member of the « groupe d'experts de l'AFNOR pour le language C++ ».

The geometricaproject maintains on its web site a collection of comprehensive sheets about the subjects presented in this report, as well as downloadable softwares. A surface reconstruction service is also available (see section ).

- In the «Master STIC» of Nice Sophia-Antipolis University Olivier Devillers chairs the second year research speciality : «Image et géométrie pour le multimédia et la modélisation du vivant»

- Olivier Devillers is professor «Chargé de cours» at École Polytechnique.

- Tsinghua University, Beijing, Advanced Computational Geometry, Jean-Daniel Boissonnat (16h)

- Ecole Centrale Paris, Introduction to Computational Structural Biology (F. Cazals - 9h, M. Nilges (Inst. Pasteur) - 6h)

- MPRI (Master Parisien de Recherches Informatiques) (2005-2006), Cours de 2ième annee , Géométrie algorithmique : Triangulations, maillages et modélisation géométrique, Jean-Daniel Boissonnat and Mariette Yvinec (24h).

- École Polytechnique (Palaiseau), Computational Geometry (2005-2006), 40h (O. Devillers).

- École Polytechnique (Palaiseau), Java programming (2005-2006), exam (O. Devillers).

- ISIA (Sophia-Antipolis), Computational Geometry (2005-2006), 10h (O. Devillers).

- Maîtrise Informatique (Nice), Computational Geometry, 24h (O. Devillers 12h, M. Pouget 12h).

- Master de Mathématiques, options Formes (Nice) (2004-2005) Triangulations, maillages et modélisation géométrique, Jean-Daniel Boissonnat (15h), Mariette Yvinec (15h).

- Master STIC-IGMMV-ISI (Sophia-Antipolis), Algorithmic frameworks for geometry (2005-2006), 15h (O. Devillers).

- Master STIC-IGMMV-ISI (Sophia-Antipolis), Implementing computational geometry (2005-2006), 20h (P. Alliez).

- Master STIC-IGMMV (Sophia-Antipolis), Surfaces and meshes (2004-2005), (F. Cazals - 9h, P. Alliez - 6h).

- Master IVR - Grenoble, Maillages et Surfaces, (F. Cazals - 6h; D. Attali (CNRS) - 6h).

- Master MIGS - Dijon, Maillages et surfaces (P. Alliez - 6h), Robustesse des algorithmes géométriques (S. Pion - 6h), Algorithmes en biologie structurale (F. Cazals - 6h).

Internship proposals can be seen on the web at http://www-sop.inria.fr/prisme/Stages/

- Daniel Loreto,
*Delaunay triangulation using swap*. MIT Internship.

- Chinmay Karande,
*Partial geometric shape matching using product graphs*IIT Bombay.

- Sébastien Loriot,
*Exact arrangement of circles on a sphere. Applications to molecular modeling*, Univ. of Dijon.

- Julien Hazebrouck,
*Curved kernel for*
cgal
*, application to VLSI*, ESSI.

- Nicolas Carrez,
*Utilisation de*
cgal
*dans le logiciel de dessin Ipe*, Supelec Metz.

- Pooran Memari,
*Calcul d'axes médians stables*, Ecole Polytechnique.

- Trung Nguyen,
*Geometric optimization for the conception of telescopes*, ENS Cachan.

- Luca Castelli,
*Compression et entropie d'objets pour la synthèse d'images*en cotutelle avec l'École Polytechnique.

- Christophe Delage,
*Non affine Voronoi diagrams*, ENS-Lyon.

- Sébastien Loriot,
*Modélisation mathématique, calcul et classification de poches d'arrimage de médicament sur les protéines*, université de Bourgogne.

- Abdelkrim Mebarki,
*Structures de données compactes pour la géométrie*, université de Nice-Sophia Antipolis.

- Thanh-Trung Nguyen,
*Geometric Optimization for the Conception of Telescopes*.

- Laurent Rineau,
*Maillages tétraédriques*, Université de Paris VI.

- Marie Samozino,
*Filtrage, simplification et représentation multirésolution d'objets géométriques reconstruits*, Université de Nice-Sophia Antipolis.

- Camille Wormser,
*Maillages et diagrammes anisotropes*.

- Thomas Lewiner,
*Computational Topology and Applications in Molecular Modeling*, École Polytechnique.

- Steve Oudot,
*Sampling and Meshing Surfaces wit Guarantees*, École Polytechnique.

- Marc Pouget,
*Geometry of surfaces: from the estimation of local differential quantities to the robust extraction of global differential features*, université de Nice-Sophia Antipolis. Defense date: 12/02/05.

Members of the project have presented all published articles at conferences. The reader can refer to the bibliography to obtain the corresponding list. We list below all other talks given in seminars or summer schools.

- «Geometry and topology for structural molecular modeling: erroneous judgments and (moderately) high hopes», Workshop
*Bioinformatics: algorithms, structures and statistics*, LIX, Ecole Polytechnique, F. Cazals.

- «Contacts in macro-molecular assemblies», Workshop
*Visualization of Large Biomolecular Complexes*, La Jolla, USA, F. Cazals.

- «Ridges and umbilics of polynomial parametric surfaces»,
*Real Algebra, Quadratic Forms and Model Theory; Algorithms and Applications*, Paris, France, M.Pouget.

- «Codage et représentations compactes en géométrie»,
*séminaire LORIA*, Nancy, O. Devillers.

- «Géométrie algorithmique, arithmétique et algèbre»,
*Journées Nationales de Calcul Formel*, CIRM, Luminy, S. Pion.

- « cgal, a computational geometry algorithms library», Summer School on Open Software for Algebraic and Geometric Computations, Sophia Antipolis, September 2005, M. Yvinec.

- «20 ans de recherche sur la reconstruction de surfaces»,
*Journée Recherche Industrie*, INRIA Rocquencourt, J.-D. Boissonnat.

-«Meshing and reconstructing surfaces», Institut Français du Pétrole, March 2005, J.-D. Boissonnat.

-«Blind approximation of surfaces»,
*Congrès de la SMAI*, Evian, May 2005, J.-D. Boissonnat.

- «Surface Learning by Probing»,
*Dagstuhl Seminar on Computational Geometry*, Dagstuhl, March 2005, J.-D. Boissonnat.

-«Voronoi Diagrams and Surface Approximation», Laboratoire d'Informatique Fondamentale, Université de Méditerranée, September 2005. J.-D. Boissonnat.

-«Surface mesh generation by Delaunay refinement», LIAMA, Pékin, October 2005. J.-D. Boissonnat.

- «Centroidal Voronoi Tessellations», SIAM annual meeting, New Orleans, July 2005. P. Alliez.

- «Optimization Techniques for Meshing», SIAM geometric design, Phoenix, October 2005. P. Alliez.

- «Optimization Techniques for Meshing», University of Aachen, Germany, April 2005. P. Alliez.

- «Geometry Compression, a Survey», EUSIPCO, Antalya, Turkey, September 2005. P. Alliez.

- «Persistance topologique», Journées de Géométrie algorithmique, Saint-Pierre de Chartreuse, Janvier 2005. D. Cohen-Steiner.

http://www-sop.inria.fr/prisme/seminaire/

The geometricaseminar featured presentations from the following visiting scientists:

M. Nilges, Institut Pasteur, France

B. Lévy, LORIA, France

Y. Babenko, Vanderbilt University, USA

J-P. Pons, INRIA Sophia, France

T. Lewiner, INRIA Sophia and PUC - Rio de Janeiro, Brazil

J-M. Schlenker, Univ. Paul Sabatier, Toulouse, France

M. Sanner, The Scripps Research Institute, La Jolla, USA

R. Marroquim, A.L. Coimbra Institute, Rio de Janeiro, Brazil

A. Ressayre, Univ. Paris-Sud / INRA

- D. Cohen-Steiner visited Duke University (USA) for one month.

- S. Pion visited the New-York University for four weeks.

- J.-D. Boissonnat visited Tsinghua University (Beijing) for two weeks.

geometricahas hosted three scientists:

- K. Saetzler, University of Ulster (August-September 2005).

- G. Vegter, University of Groningen (May 2005).

- C. Rössl, MPII Saarbrucken, Germany (December 2005).