André Cerezo

Shortest paths of bounded curvature in the plane

Given two oriented points in the plane, we determine and compute the shortest paths of bounded curvature joining them. This problem has been solved recently by Dubins in the no-cusp case, and by Reeds and Shepp otherwise. We propose a new solution based on the minimum principle of Pontryagin. Our approach simplifies the proofs and makes clear the global or local nature of the results.

An algorithm for constructing the convex hull of a set of spheres in dimension d

Abstract We present an algorithm which computes the convex hull of a set of n spheres in dimension d in time O (n ⌜ 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 homothetic convex objects of Ed. If the complexity of each object is constant, the time needed in the worst case is O (n ⌜ d 2 ri + n log n) .

OUTPUT SENSITIVE CONSTRUCTION OF THE DELAUNAY TRIANGULATION OF POINTS LYING IN TWO PLANES

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).

Shortest plane paths with bounded derivative of the curvature

Abstract We study the shortest plane paths joining two given points with given tangent angles and curvatures along which the tangent angle and the curvature are continuous and the derivative of the curvature is bounded. At a point where such a path is of class C3, it must be locally a piece of a clothoid or a line segment (up to an isometry, a clothoid is given by Fresnels integrals x(t) = ∝0t cos τ2 dτ, y(t) = ∝0t sin τ2 dτ, where t is the arc length). We prove that if the distance between its initial and final points is large enough, then a generic shortest path contains infinitely many switching points.

On a Group of Unitary Deformations

Abstract We develop in this work an idea initially due to J.M.Levy-Leblond ([10]) for deforming through explicit homotopies, the Fourier transform into identity, the convolution product into pointwise product, differential operators into others, et cetera, and we express here our best thanks to him. The main tools to achieve this are the Hermite expansion of any tempered distribution, due to L.Schwartz ([6]) but not much used since, and Mahler’s formula on Hermite functions, which allows us an explicit computation of the kernels of all the operators involved.