The Limit of Lp Voronoi Diagrams as p→0 is the Bounding-Box-Area Voronoi Diagram
We consider the Voronoi diagram of points in the real plane when the distance between two points a and b is given by Lp(a-b) where Lp((x,y))=(|x|p+|y|p)1/p. We prove that the Voronoi diagram has a limit as p converges to zero from above or from below: it is the diagram that corresponds to the distan...
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| Published in | Discrete & computational geometry Vol. 72; no. 3; pp. 1284 - 1303 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
New York
Springer Nature B.V
01.10.2024
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| Online Access | Get full text |
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| Abstract | We consider the Voronoi diagram of points in the real plane when the distance between two points a and b is given by Lp(a-b) where Lp((x,y))=(|x|p+|y|p)1/p. We prove that the Voronoi diagram has a limit as p converges to zero from above or from below: it is the diagram that corresponds to the distance function L∗((x,y))=|xy|. In this diagram, the bisector of two points in general position consists of a line and two branches of a hyperbola that split the plane into three faces per point. We propose to name L∗ as defined above the geometric L0 distance. |
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| AbstractList | We consider the Voronoi diagram of points in the real plane when the distance between two points a and b is given by Lp(a-b) where Lp((x,y))=(|x|p+|y|p)1/p. We prove that the Voronoi diagram has a limit as p converges to zero from above or from below: it is the diagram that corresponds to the distance function L∗((x,y))=|xy|. In this diagram, the bisector of two points in general position consists of a line and two branches of a hyperbola that split the plane into three faces per point. We propose to name L∗ as defined above the geometric L0 distance. |
| Author | Haverkort, Herman Klein, Rolf |
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| ContentType | Journal Article |
| Copyright | The Author(s) 2023. This work is published under http://creativecommons.org/licenses/by/4.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License. |
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| Snippet | We consider the Voronoi diagram of points in the real plane when the distance between two points a and b is given by Lp(a-b) where Lp((x,y))=(|x|p+|y|p)1/p. We... |
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| Title | The Limit of Lp Voronoi Diagrams as p→0 is the Bounding-Box-Area Voronoi Diagram |
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