Algorithm 772

We review the use of Voronoi tessellations for grid generation, especially on the whole sphere or in regions on the sphere. Voronoi tessellations and the corresponding Delaunay tessellations in regions and surfaces on Euclidean space are defined and properties they possess that make them well-suited for grid generation purposes are discussed, as are algorithms for their construction. This is followed by a more detailed look at one very special type of Voronoi tessellation, the centroidal Voronoi tessellation (CVT). After defining them, discussing some of their properties, and presenting algorithms for their construction, we illustrate the use of CVTs for producing both quasi-uniform and variable resolution meshes in the plane and on the sphere. Finally, we briefly discuss the computational solution of model equations based on CVTs on the sphere.

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“…In particular, spherical Voronoi tessellation and Delaunay triangulation and related algorithms are developed in (Renka, 1997).…”

Section: Definitions and Propertiesmentioning

confidence: 99%

“…There also exist software packages that may be used for Voronoi tessellation construction. For example, on the sphere, there is the STRIPACK package (Renka, 1997).…”

Section: Determine the Centroids (Or Constrained Centroids) With Resmentioning

confidence: 99%

Voronoi Tessellations and Their Application to Climate and Global Modeling

Ringler

Gunzburger

2011

Lecture Notes in Computational Science and Engineering

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“…For spherical space, we use STRIPACK [Renka, 1997] to compute the Voronoi diagram on a sphere. For hyperbolic space, as discussed above, computing a hyperbolic Voronoi diagram in Klein disk can be easily accomplished by computing a power diagram in Euclidean space.…”

Section: Implementation Detailsmentioning

confidence: 99%

Centroidal Voronoi tessellation in universal covering space of manifold surfaces

Rong

Jin

Shu

et al. 2011

Computer Aided Geometric Design

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The centroidal Voronoi tessellation (CVT) has found versatile applications in geometric modeling, computer graphics, and visualization, etc. In this paper, we first extend the concept of CVT from Euclidean space to spherical space and hyperbolic space, and then combine all of them into a unified framework -the CVT in universal covering space. The novel spherical and hyperbolic CVT energy functions are defined, and the relationship between minimizing the energy and the CVT is proved. We also show by our experimental results that both spherical and hyperbolic CVTs have the similar property as their Euclidean counterpart where the sites are uniformly distributed with respect to given density values. As an example of the application, we utilize the CVT in universal covering space to compute uniform partitions and high-quality remeshing results for genus-0, genus-1, and high-genus (genus>1) surfaces.

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“…Such discretization is a standard solution, which one may find in convex geometry [4] through Delaunay triangulation [1,8,11]. Regarding to the global and local conditions we take into consideration a fact, that we cannot generate polygons in R 2 which overlap neighbour polygons.…”

Section: Introductionmentioning

confidence: 99%

“…We use Delaunay triangulation [1,8,11] that to find neighbourhood of point P i defined by several points P j(i,1) . .…”

Section: Introductionmentioning

confidence: 99%