This work studies path planning in two-dimensional space, in the presence of polygonal obstacles. We specifically address the problem of building a roadmap graph, that is, an abstract representation of all the paths that can potentially be followed around a given set of obstacles. Our solution consists in an original refinement algorithm for constrained Delaunay triangulations, aimed at generat...

We propose the use of the Gabriel graph for the exploratory analysis of potentially high dimensional labeled data. Gabriel graph is a subgraph of the Delaunay triangulation, which connects two data points vi and vj for which there is no other point vk inside the open ball with diameter [vivj ]. If all the Gabriel neighbors of a datum have a different class than its own, this datum is said to be...

Given a finite set of points S ⊆ R, we define a proximity graph called as shape-hull graph(SHG(S)) that contains all Gabriel edges and a few non-Gabriel edges of Delaunay triangulation of S. For any S, SHG(S) is topologically regular with its boundary (referred to as shape-hull(SH)) homeomorphic to a simple closed curve. We introduce the concept of divergent concavity for simple, closed, planar...

Many image segmentation approaches rely upon or are enhanced by using spatial relationship information between image regions and their object correspondences. Spatial relationships are usually captured in terms of relative neighborhood graphs such as the Delaunay graph. Neighborhood graphs capture information about which objects are close to each other in the plane or in space but may not captu...

β-skeletons, a prominent member of the neighborhood graph family, have interesting geometric properties and various applications ranging from geographic networks to archeology. This paper focuses on developing a new, more general than the present one, definition of β-skeletons based only on the distance criterion. It allows us to consider them in many different cases, e.g. for weighted graphs o...

We present a deterministic local routing scheme that is guaranteed to find a path between any pair of vertices in a halfθ6-graph whose length is at most 5/ √ 3 = 2.886... times the Euclidean distance between the pair of vertices. The half-θ6graph is identical to the Delaunay triangulation where the empty region is an equilateral triangle. Moreover, we show that no local routing scheme can achie...

We describe an algorithm to construct an intrinsic Delaunay triangulation of a smooth closed submanifold of Euclidean space. Using results established in a companion paper on the stability of Delaunay triangulations on δ-generic point sets, we establish sampling criteria which ensure that the intrinsic Delaunay complex coincides with the restricted Delaunay complex and also with the recently in...

We introduce a parametrized notion of genericity for Delaunay triangulations which, in particular, implies that the Delaunay simplices of δ-generic point sets are thick. Equipped with this notion, we study the stability of Delaunay triangulations under perturbations of the metric and of the vertex positions. We quantify the magnitude of the perturbations under which the Delaunay triangulation r...