# Karlsruhe metric

Jump to: navigation, search

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

${\displaystyle d_{k}(p_{1},p_{2})={\begin{cases}\min(r_{1},r_{2})\cdot \delta (p_{1},p_{2})+|r_{1}-r_{2}|,&{\text{if }}0\leq \delta (p_{1},p_{2})\leq 2\\r_{1}+r_{2},&{\text{otherwise}}\end{cases}}}$

where ${\displaystyle (r_{i},\varphi _{i})}$ are the polar coordinates of ${\displaystyle p_{i}}$ and ${\displaystyle \delta (p_{1},p_{2})=\min(|\varphi _{1}-\varphi _{2}|,2\pi -|\varphi _{1}-\varphi _{2}|)}$ is the angular distance between the two points.