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(p₁ 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(p₁), . . . , 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.

Reference

[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.