Karlsruhe metric

In metric geometry, the Karlsruhe metric is a measure of distance 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.^[1] The Karlsruhe distance between two points [math]\displaystyle{ d_k(p_1,p_2) }[/math] is given as

[math]\displaystyle{ d_k(p_1,p_2)= \begin{cases} \min(r_1,r_2) \cdot \delta(p_1,p_2) +|r_1-r_2|,&\text{if } 0\leq \delta(p_1,p_2)\leq 2\\ r_1+r_2,&\text{otherwise} \end{cases} }[/math]

where [math]\displaystyle{ (r_i,\varphi_i) }[/math] are the polar coordinates of [math]\displaystyle{ p_i }[/math] and [math]\displaystyle{ \delta(p_1,p_2)=\min(|\varphi_1-\varphi_2|,2\pi-|\varphi_1-\varphi_2|) }[/math] is the angular distance between the two points.

See also

• Metric (mathematics)
• Manhattan distance
• Hamming distance

Notes

External links

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

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