DEVILLERS , Olivier

A good sample is a point set such that any ball of radius $\epsilon$ contains a constant number of points. The Delaunay triangulation of a good sample is proved to have linear size, unfortunately this is not enough to ensure a good time complexity of the randomized incremental construction of the Delaunay triangulation. In this paper we prove...

Though Delaunay triangulations are very well known geometric data structures, the problem of the robust removal of a vertex in a three-dimensional Delaunay triangulation is still a problem in practice. We propose a simple method that allows to remove any vertex even when the points are in very degenerate configurations. The solution is...

We present a simple technique for analyzing the size of geometric hypergraphs dened by random point sets. As an application we obtain upper and lower bounds on the smoothed number of faces of the convex hull under Euclidean and Gaussian noise and related results.

We are interested in computing effectively cylinders through 5 points, and in other problems involved in metrology. In particular, we consider the cylinders through 4 points with a fix radius and with extremal radius. For these different problems, we give bounds on the number of solutions and exemples show that these bounds are optimal....

A simple method to produce a random order type is to take the order type of a random point set. We conjecture that many probabilitydistributions on order types defined in this way are heavily concentrated and therefore sample inefficiently the space of order types. We present two results on this question. First, we study experimentally the bias...

We are interested in computing effectively cylinders through 5 points, and in other problems involved in metrology. In particular, we consider the cylinders through 4 points with a fix radius and with extremal radius. For these different problems, we give bounds on the number of solutions and exemples show that these bounds are optimal....

We say that a simplicial complex is shrinkable if there exists a sequence of admissible edge contractions that reduces the complex to a single vertex. We prove that it is NP-complete to decide whether a (two-dimensional) simplicial complex is shrinkable. Along the way, we describe examples of contractible complexes that are not shrinkable.

In this paper, we propose an algorithm to compute the Delaunay triangulation of a set of n points in 3-dimensional space when the points lie in 2 planes. The algorithm is output-sensitive and optimal with respect to the input and the output sizes. Its time complexity is O(n log n+t), where t is the size of the output, and the extra storage is O(n).

The randomized incremental construction (RIC) for building geometric data structures has been analyzed extensively, from the point of view of worst-case distributions. In many practical situations however, we have to face nicer distributions. A natural question that arises is: do the usual RIC algorithms automatically adapt when the point...

A cover for a family $\mathcal{F}$ of sets in the plane is a set into which every set in $\mathcal{F}$ can be isometrically moved. We are interested in the convex cover of smallest area for a given family of triangles. Park and Cheong conjectured that any family of triangles of bounded diameter has a smallest convex cover that is itself a...

Randomized incremental construction (RIC) is one of the most important paradigms for building geometric data structures. Clarkson and Shor developed a general theory that led to numerous algorithms that are both simple and efficient in theory and in practice. Randomized incremental constructions are most of the time space and time optimal in...

Randomized incremental construction(RIC) is one of the most important paradigms for buildinggeometric data structures. Clarkson and Shor developed a general theory that led to numerous algorithms that are both simple and efficient in theory and in practice.Randomized incremental constructions are most of the time space and time optimal in the...

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A cover for a family F of sets in the plane is a set into which every set in F can be isometrically moved. We are interested in the convex cover of smallest area for a given family of triangles. Park and Cheong conjectured that any family of triangles of bounded diameter has a smallest convex cover that is itself a triangle. The conjecture is...

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Randomized incremental construction (RIC) is one of the most important paradigms for building geometric data structures. Clarkson and Shor developed a general theory that led to numerous algorithms that are both simple and efficient in theory and in practice. Randomized incremental constructions are usually space-optimal and time-optimal in the...

We present two lower bounds that hold with high probability for random point sets. We first give a new, and elementary, proof that the classical models of random point sets (uniform in a smooth convex body, uniform in a polygon, Gaussian) have a superconstant number of extreme points with high probability. We next prove that any algorithm that...

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We present an algorithm which computes the convex hull of a set of spheres in dimension d in time O(n^ceil(d/2)+n log n). It is worst case optimal in three dimensions and in even dimensions. The same method can also be used to compute the convex hull of a set of n homethetic objects of Ed. If the complexity of each object is constant, the time...

We propose a method to evaluate signs of $2\times 2$ and $3\times 3$ determinants with $b$-bit integer entries using only $b$ and $(b+1)$-bit arithmetic respectively. This algorithm has numerous applications in geometric computation and provides a general and practical approach to robustness. The algorithm has been implemented and experimental...

We study the following problem: Given $k$ paths that share the same vertex set, is there a simultaneous geometric embedding of these paths such that each individual drawing is monotone in some direction? We prove that for any dimension $d\geq 2$, there is a set of $d + 1$ paths that does not admit a monotone simultaneous geometric embedding.

In most layered additive manufacturing processes, a tool solidifies or deposits material while following pre-planned trajectories to form solid beads. Many interesting problems arise in this context, among which one concerns the planning of trajectories for filling a planar shape as densely as possible. This is the problem we tackle in the...

We study the following problem: Given $k$ paths that share the same vertex set, is there a simultaneous geometric embedding of these paths such that each individual drawing is monotone in some direction? We prove that for any dimension $d\geq 2$, there is a set of $d + 1$ paths that does not admit a monotone simultaneous geometric embedding.