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 ${\displaystyle d_k(p_1,p_2)}$ is given as

${\displaystyle d_k(p_1,p_2)=\{\begin{array}{ll}\min (r_1,r_2)\cdot \delta (p_1,p_2)+|r_1-r_2|,& \text{if }0\le \delta (p_1,p_2)\le 2\\ r_1+r_2,& \text{otherwise}\end{array}}$

where ${\displaystyle (r_i,{\phi }_i)}$ are the polar coordinates of ${\displaystyle p_i}$ and ${\displaystyle \delta (p_1,p_2)=\min (|{\phi }_1-{\phi }_2|,2\pi -|{\phi }_1-{\phi }_2|)}$ is the angular distance between the two points.

• Metric (mathematics)
• Manhattan distance
• Hamming distance

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

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