Karlsruhe metric

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In metric geometry, the Karlsruhe metric (the name alludes to the layout of the city of Karlsruhe), also called Moscow metric, is a measure of distance that assumes travel is only possible along radial streets and along circular avenues around the center.[1]

The Karlsruhe distance between two points d_k(p_1,p_2) is given as


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}

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

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