This paper proposes a new method for isotropic remeshing of triangulated surface meshes. Given a triangulated surface mesh to be resampled and a user-specified density function defined over it, we first distribute the desired number of samples by generalizing error diffusion, commonly used in image halftoning, to work directly on mesh triangles and feature edges. We then use the resulting sampling as an initial configuration for building a weighted centroidal Voronoi diagram in a conformal parameter space, where the specified density function is used for weighting. We finally create the mesh by lifting the corresponding constrained Delaunay triangulation from parameter space. A precise control over the sampling is obtained through a flexible design of the density function, the latter being possibly low-pass filtered to obtain a smoother gradation. We demonstrate the versatility of our approach through various remeshing examples.

We describe an algorithm which, for any piecewise linear complex (PLC) in 3D, builds a Delaunay triangulation conforming to this PLC. The algorithm has been implemented, and yields in practice a relatively small number of Steiner points due to the fact that it adapts to the local geometry of the PLC. It is, to our knowledge, the first practical algorithm devoted to this problem.

Let M be an orientable surface without boundary. A cycle on M is splitting if it has no self-intersections and it partitions M into two components, neither homeomorphic to a disk. In other words, splitting cycles are simple, separating, and non-contractible. We prove that finding the shortest splitting cycle on a combinatorial surface is NP-hard but fixed-parameter tractable with respect to the surface genus. Specifically, we describe an algorithm to compute the shortest splitting cycle in g O(g) n 2 log n time.

Abstract. Let D be a weighted directed graph cellularly embedded in a surface of genus g, orientable or not, possibly with boundary. We describe algorithms to compute shortest noncontractible and shortest surface non-separating cycles in D, generalizing previous results that dealt with undirected graphs.Our first algorithm computes such cycles in O(n 2 log n) time, where n is the total number of vertices and edges of D, thus matching the complexity of the best general algorithm in the undirected case. It revisits and extends Thomassen's 3-path condition; the technique applies to other families of cycles as well.We also provide more efficient algorithms in special cases, such as graphs with small genus or bounded treewidth, using a divide-and-conquer technique that simplifies the graph while preserving the topological properties of its cycles. Finally, we give an efficient output-sensitive algorithm, whose running time depends on the length of the shortest non-contractible or non-separating cycle.

Let M be an orientable combinatorial surface without boundary. A cycle on M is splitting if it has no self-intersections and it partitions M into two components, neither of which is homeomorphic to a disk. In other words, splitting cycles are simple, separating, and non-contractible. We prove that finding the shortest splitting cycle on a combinatorial surface is NP-hard but fixed-parameter tractable with respect to the surface genus. Specifically, we describe an algorithm to compute the shortest splitting cycle in g O(g) n log n time.

The Fréchet distance between two curves in the plane is the minimum length of a leash that allows a dog and its owner to walk along their respective curves, from one end to the other, without backtracking. We propose a natural extension of Fréchet distance to more general metric spaces, which requires the leash itself to move continuously over time. For example, for curves in the punctured plane, the leash cannot pass through or jump over the obstacles ("trees"). We describe a polynomial-time algorithm to compute the homotopic Fréchet distance between two given polygonal curves in the plane minus a given set of polygonal obstacles.

A triangulation of a surface is irreducible if no edge can be contracted to produce a triangulation of the same surface. In this paper, we investigate irreducible triangulations of surfaces with boundary. We prove that the number of vertices of an irreducible triangulation of a (possibly non-orientable) surface of genus g ≥ 0 with b ≥ 0 boundary components is O (g + b). So far, the result was known only for surfaces without boundary (b = 0). While our technique yields a worse constant in the O(.) notation, the present proof is elementary, and simpler than the previous ones in the case of surfaces without boundary.

This paper is concerned with applications of Tutte's barycentric embedding theorem (Proc. London Math. Soc. 13 (1963), 743-768). It presents a method for building isotopies of triangulations in the plane, based on Tutte's theorem and the computation of equilibrium stresses of graphs by Maxwell-Cremona's theorem; it also provides a counterexample showing that the analogue of Tutte's theorem in dimension 3 is false.