Geodesic-Preserving Polygon Simplification

Polygons are a paramount data structure in computational geometry. While the complexity of many algorithms on simple polygons or polygons with holes depends on the size of the input polygon, the intrinsic complexity of the problems these algorithms solve is often related to the reflex vertices of the polygon. In this paper, we give an easy-to-describe linear-time method to replace an input polygon $\mathcal{P}$ by a polygon $\mathcal{P}'$ such that (1) $\mathcal{P}'$ contains $\mathcal{P}$, (2) $\mathcal{P}'$ has its reflex vertices at the same positions as $\mathcal{P}$, and (3) the number of vertices of $\mathcal{P}'$ is linear in the number of reflex vertices. Since the solutions of numerous problems on polygons (including shortest paths, geodesic hulls, separating point sets, and Voronoi diagrams) are equivalent for both $\mathcal{P}$ and $\mathcal{P}'$, our algorithm can be used as a preprocessing step for several algorithms and makes their running time dependent on the number of reflex vertices rather than on the size of $\mathcal{P}$

On the statistics of area size in two-dimensional thick Voronoi Diagrams

Cells of Voronoi diagrams in two dimensions are usually considered as having edges of zero width. However, this is not the case in several experimental situations in which the thickness of the edges of the cells is relatively large. In this paper, the concept of a thick Voronoi tessellation, that is with edges of non-zero width, is introduced and the the statistics of cell areas, as thickness changes, are analyzed.Comment: 14 pages, 9 figure

A simple and efficient model for mesoscale solidification simulation of globular grain structures

A simple model for the solidification of globular grains in metallic alloys is presented. Based on the Voronoi diagram of the nuclei centers, it accounts for the curvature of the grains near triple junctions. The predictions of this model are close to those of more refined approaches such as the phase field method, but with a computation cost decreased by several orders of magnitude. Therefore, this model is ideally suited for granular simulations linking the behavior of individual grains to macroscopic properties of the material