Discrete Laplace operators are ubiquitous in applications spanning geometric modeling to simulation. For robustness and efficiency, many applications require discrete operators that retain key structural properties inherent to the continuous setting. Building on the smooth setting, we present a set of natural properties for discrete Laplace operators for triangular surface meshes. We prove an important theoretical limitation: discrete Laplacians cannot satisfy all natural properties; retroactively, this explains the diversity of existing discrete Laplace operators. Finally, we present a family of operators that includes and extends well-known and widely-used operators.

We show that, for any collection H of n hyperplanes in ! 4 , the combinatorial complexity of the vertical decomposition of the arrangement A(H) of H is O(n 4 log n). The proof relies on properties of superimposed convex subdivisions of 3-space, and we also derive some other results concerning them.

: We consider generalization of the Voronoi diagram - power diagram, constructed with respect to a Poisson processes with i.i.d. marks (weights). We give first moment of the volume distribution of a typical cell, the probability that a cell is empty, the mean length of a typical edge in the planar case and other geometrical characteristics of this tesselation. AMS 1991 Subject Classification: 60D05, 52C17, 60G55 Key-words: Power diagrams, Voronoi tesselation, Poisson process, Planar tesselations (R'esum'e : tsvp) e-mail: zouev@sophia.inria.fr Unite de recherche INRIA Sophia-Antipolis 2004 route des Lucioles, BP 93, 06902 SOPHIA-ANTIPOLIS Cedex (France) Telephone : (33) 93 65 77 77 -- Telecopie : (33) 93 65 77 65 Tesselations poissoniennes de la puissance R'esum'e : On consid`ere le diagramme de puissance. Il s'agit d'une g'en'eralisation de la tess'elation de Voronoi construite par rapport `a un processus poissonien marqu'e par des poids i.i.d. Nous donnons les premiers moments ...

We propose four simple algorithms for routing on planar graphs using virtual coordinates. These algorithms are superior to existing algorithms in that they are oblivious, work also for non-triangular graphs, and their virtual coordinates are easy to construct.

A parallelotope is a polytope whose translation copies fill space without gaps and intersections by interior points. Voronoi conjectured that each parallelotope is an affine image of the Dirichlet domain of a lattice, which is a Voronoi polytope. We give several properties of a parallelotope and prove that each of them is equivalent to it is an affine image of a Voronoi polytope. 1