Generalizing CGAL Periodic Delaunay Triangulations

Osang, Georg; Rouxel-Labbé, Mael; Teillaud, Monique

doi:10.4230/lipics.esa.2020.75

Cited by 4 publications

(2 citation statements)

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“…Voronoi diagrams and Delaunay triangulations are made from grain centers, including the eight copies of these centers that are used for obeying the periodic boundary conditions. A more efficient procedure to construct Voronoi diagrams and Delaunay triangulations with periodic boundaries might be available in the near future [52].…”

Section: Methods For Obtaining Thresholdsmentioning

confidence: 99%

Formation and morphology of closed and porous films grown from grains seeded on substrates: Two-dimensional simulations

Janssens,

Vázquez-Cortés,

Fried

2021

Preprint

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Two-dimensional simulations are used to explore topological transitions that occur during the formation of films grown from grains that are seeded on substrates. This is done for a relatively large range of the initial value Φ s of the grain surface fraction Φ. The morphology of porous films is captured at the transition when grains connect to form a one-component network using newly developed raster-free algorithms that combine computational geometry and network theory. Further insight on the morphology of porous films and their suspended counterparts is obtained by studying the pore surface fraction Φ p , the pore over grain ratio, the pore area distribution, and the contribution of pores of certain chosen areas to Φ p . Pinhole survival is evaluated at the transition when film closure occurs using survival function estimates. The morphology of closed films (Φ = 1) is also characterized and is quantified by measuring grain areas and perimeters. The majority of investigated quantities are found to depend sensitively on Φ s and the long-time persistence of pinholes exhibits critical behavior as a function of Φ s . In addition to providing guidelines for designing effective processes for manufacturing thin films and suspended porous films with tailored properties, this work may advance the understanding of continuum percolation theory.

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Section: Methods For Obtaining Thresholdsmentioning

confidence: 99%

Formation and morphology of closed and porous films grown from grains seeded on substrates: Two-dimensional simulations

Janssens,

Vázquez-Cortés,

Fried

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Other practical approaches for (flat) quotient spaces start by computing in a finite-sheeted covering space [14] or in the universal covering space [34], thus requiring the duplication of some input points. In contrast to this approach, our algorithm proceeds directly on the surface, thus circumventing the need for duplicating any input points.…”

Section: Introductionmentioning

confidence: 99%

Delaunay triangulations of generalized Bolza surfaces

Ebbens¹,

Iordanov²,

Teillaud³

et al. 2021

Preprint

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The Bolza surface can be seen as the quotient of the hyperbolic plane, represented by the Poincaré disk model, under the action of the group generated by the hyperbolic isometries identifying opposite sides of a regular octagon centered at the origin. We consider generalized Bolza surfaces Mg, where the octagon is replaced by a regular 4g-gon, leading to a genus g surface. We propose an extension of Bowyer's algorithm to these surfaces. In particular, we compute the value of the systole of Mg. We also propose algorithms computing small sets of points on Mg that are used to initialize Bowyer's algorithm.

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Representing Infinite Periodic Hyperbolic Delaunay Triangulations Using Finitely Many Dirichlet Domains

Despré,

Kolbe,

Teillaud

2024

Discrete Comput Geom

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The Delaunay triangulation of a set of points P on a hyperbolic surface is the projection of the Delaunay triangulation of the set $$\widetilde{P}$$ P ~ of lifted points in the hyperbolic plane. Since $$\widetilde{P}$$ P ~ is infinite, the algorithms to compute Delaunay triangulations in the plane do not generalize naturally. Using a Dirichlet domain, we exhibit a finite set of points that captures the full triangulation. We prove that an edge of a Delaunay triangulation has a combinatorial length (a notion we define in the paper) smaller than $$12g-6$$ 12 g - 6 with respect to a Dirichlet domain. To achieve this, we introduce new tools, of intrinsic interest, that capture the properties of length-minimizing curves in the context of closed curves. We then use these to derive structural results on Delaunay triangulations and exhibit certain distance minimizing properties of both the edges of a Delaunay triangulation and of a Dirichlet domain. The bounds produced in this paper depend only on the topology of the surface. They provide mathematical foundations for hyperbolic analogs of the algorithms to compute periodic Delaunay triangulations in Euclidean space.

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Generalizing CGAL Periodic Delaunay Triangulations

Cited by 4 publications

References 0 publications

Formation and morphology of closed and porous films grown from grains seeded on substrates: Two-dimensional simulations

Formation and morphology of closed and porous films grown from grains seeded on substrates: Two-dimensional simulations

Delaunay triangulations of generalized Bolza surfaces

Representing Infinite Periodic Hyperbolic Delaunay Triangulations Using Finitely Many Dirichlet Domains

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