Karlsruhe metric

 ( Metric Spaces ) 

In metric geometry, the Karlsruhe metric is a metric space that assumes travel is only possible along rays through the origin and circular arcs centered at the origin. The name alludes to the layout of the city of Karlsruhe, which has radial streets and circular avenues around a central point. This metric is also called Moscow metric. Karlsruhe-Metric Voronoi Diagram

In this metric, there are two types of shortest paths. One possibility, when the two points are on nearby rays, combines a circular arc through the nearer to the origin of the two points and a segment of a ray through the farther of the two points. Alternatively, for points on rays that are nearly opposite, it is shorter to follow one ray all the way to the origin and then follow the other ray back out. Therefore, the Karlsruhe distance between two points $d\_k(p\_1,p\_2)$ is the minimum of the two lengths that would be obtained for these two types of path. That is, it equals where $(r\_i,\backslash varphi\_i)$ are the polar coordinates of $p\_i$ and $\backslash delta(p\_1,p\_2)=\backslash min(|\backslash varphi\_1-\backslash varphi\_2|,2\backslash pi-|\backslash varphi\_1-\backslash varphi\_2|)$ is the angular distance between the two points.

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See also

• Manhattan distance
• Hamming distance

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Notes

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External links

• Karlsruhe-metric Voronoi diagram, by Takashi Ohyama
• Karlsruhe-Metric Voronoi Diagram, by Rashid Bin Muhammad

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