In Manhattan, we can move along either North-South streets or East-West avenues. In Karlshue, we can move along either radiating streets from the center or circular avenues around the center. Specifically, let (r, θ) and (r_{i}, θ_{i}) be the polar coordinates of p and p_{i}, respectively, where 0 < θ, θ_{i}< 2π, r, r_{i} > 0, and δ(θ,θ_{i}) = min { |θ - θ_{i}|, 2π - |θ - θ_{i}|}
Then, the Karlsruhe metric or Moscow metric from p to p_{i} is defined by
min {r, r_{i}}
δ(θ,θ_{i})
+ | r - r_{i}_{ }| for 0 ≤
δ(θ,θ_{i})
< 2
d_{k}(p_{1} p_{i} ) = r + r_{i}_{
} for 2 ≤
δ(θ,θ_{i})
< π
If we apply the above metric to the bisector, the set vk = {V(p_{1}), . . . , V(p_{n})} gives generalized Voronoi diagram. This type of Voronoi diagram is known as the Karlsruhe Voronoi Diagram or simply the Karlsruhe Voronoi Diagram.
[2] R. Klein, "Abstract Voronoi diagrams and their applications Lecture Notes in Computer Science", 333 (International Workshop on Computational Geometry, Wurzburg, March 1988), Springer-Verlag, pp. 148-157.