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Hyperbolic Moroni diagrams made easy Frank Nielsen1 and Richard Nock2arXiv:0903.3287v1 cs.CG 19 Mar 20091 Cole Poly technique / Sony Computer Science Laboratories Inc Plateau, France / Tokyo, Japan
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How to fill out hyperbolic voronoi diagrams made
01
To fill out hyperbolic Voronoi diagrams, follow these steps:
02
Start with a hyperbolic space, which is a non-Euclidean space with constant negative curvature.
03
Determine the set of points, called seeds, that will serve as the centers for the Voronoi cells.
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Compute the boundaries of each Voronoi cell by finding the locus of points that are equidistant to the surrounding seeds.
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In hyperbolic space, the distance between two points is not the same as in Euclidean space. Therefore, you need to use a hyperbolic metric to calculate the distances.
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Use the hyperbolic metric to determine the distances between each seed and its neighboring seeds.
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Using these distances, define the boundaries of each Voronoi cell by finding the locus of points that are equidistant to the surrounding seeds.
08
Once the boundaries of each Voronoi cell are defined, you can fill out the cells with colors or patterns to visualize the diagram.
Who needs hyperbolic voronoi diagrams made?
01
Hyperbolic Voronoi diagrams are useful for various applications, including:
02
- Statistical modeling: Hyperbolic Voronoi diagrams can be used to model spatial distributions of data points in non-Euclidean spaces.
03
- Network analysis: These diagrams can help analyze and optimize network designs in non-Euclidean environments, such as the Internet.
04
- Spatial planning: Hyperbolic Voronoi diagrams can assist in urban planning and resource allocation in non-Euclidean spaces.
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- Computer graphics: Creating hyperbolic Voronoi diagrams can be helpful in generating visually interesting patterns and textures for digital art and animation.
06
- Geometry research: These diagrams are of interest to mathematicians and researchers studying hyperbolic geometry and its applications.
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