scholarly journals Walking in a Planar Poisson–Delaunay Triangulation: Shortcuts in the Voronoi Path

2018 ◽  
Vol 28 (03) ◽  
pp. 255-269 ◽  
Author(s):  
Olivier Devillers ◽  
Louis Noizet
Keyword(s):  
Point Process ◽  
Delaunay Triangulation ◽  
Poisson Point Process ◽  
Expected Length

Let [Formula: see text] be a planar Poisson point process of intensity [Formula: see text]. We give a new proof that the expected length of the Voronoi path between [Formula: see text] and [Formula: see text] in the Delaunay triangulation associated with [Formula: see text] is [Formula: see text] when [Formula: see text] goes to infinity; and we also prove that the variance of this length is [Formula: see text]. We investigate the length of possible shortcuts in this path, and define a shortened Voronoi path whose expected length can be expressed as an integral that is numerically evaluated to [Formula: see text]. The shortened Voronoi path has the property to be locally defined; and is shorter than the previously known locally defined paths in Delaunay triangulation such as the upper path whose expected length is [Formula: see text].

scholarly journals Stretch factor in a planar Poisson–Delaunay triangulation with a large intensity

2018 ◽  
Vol 50 (01) ◽  
pp. 35-56 ◽  
Author(s):  
Nicolas Chenavier ◽  
Olivier Devillers
Keyword(s):  
Lower Bound ◽  
Point Process ◽  
Delaunay Triangulation ◽  
Shortest Path ◽  
Upper Bound ◽  
Poisson Point Process ◽  
Large Intensity ◽  
Stretch Factor ◽  
Expected Length

Abstract Let X := X n ∪ {(0, 0), (1, 0)}, where X n is a planar Poisson point process of intensity n. We provide a first nontrivial lower bound for the distance between the expected length of the shortest path between (0, 0) and (1, 0) in the Delaunay triangulation associated with X when the intensity of X n goes to ∞. Simulations indicate that the correct value is about 1.04. We also prove that the expected length of the so-called upper path converges to 35 / 3π2, yielding an upper bound for the expected length of the smallest path.

scholarly journals On Laslett's transform for the Boolean model

2001 ◽  
Vol 33 (1) ◽  
pp. 1-5 ◽  
Author(s):  
A. D. Barbour ◽  
V. Schmidt
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Simple Proof ◽  
Poisson Point Process ◽  
Boolean Model ◽  
Random Sets ◽  
Convex Compact ◽  
Homogeneous Poisson Process ◽  
Vertical Strip ◽  
Cumulative Process

Consider the Boolean model in ℝ2, where the germs form a homogeneous Poisson point process with intensity λ and the grains are convex compact random sets. It is known (see, e.g., Cressie (1993, Section 9.5.3)) that Laslett's rule transforms the exposed tangent points of the Boolean model into a homogeneous Poisson process with the same intensity. In the present paper, we give a simple proof of this result, which is based on a martingale argument. We also consider the cumulative process of uncovered area in a vertical strip and show that a (linear) Poisson process with intensity λ can be embedded in it.

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