Editing Voronoi diagram (section)

==Illustration==
As a simple illustration, consider a group of shops in a city. Suppose we want to estimate the number of customers of a given shop. With all else being equal (price, products, quality of service, etc.), it is reasonable to assume that customers choose their preferred shop simply by distance considerations: they will go to the shop located nearest to them. In this case the Voronoi cell <math>R_k</math> of a given shop <math>P_k</math> can be used for giving a rough estimate on the number of potential customers going to this shop (which is modeled by a point in our city).

For most cities, the distance between points can be measured using the familiar
[[Euclidean distance]]:

:<math>\ell_2 = d\left[\left(a_1, a_2\right), \left(b_1, b_2\right)\right] = \sqrt{\left(a_1 - b_1\right)^2 + \left(a_2 - b_2\right)^2}</math>

or the [[Manhattan distance]]:

:<math>d\left[\left(a_1, a_2\right), \left(b_1, b_2\right)\right] = \left|a_1 - b_1\right| + \left|a_2 - b_2\right|</math>.

The corresponding Voronoi diagrams look different for different distance metrics.

{{multiple image
 | align     = center
 | direction = horizontal
 | width     = 382
 | header    = Voronoi diagrams of 20 points under two different metrics
 | header_align = center
 | image1    = Euclidean Voronoi diagram.svg
 | alt1      = Voronoi diagram under Euclidean distance
 | caption1  = [[Euclidean distance]]
 | image2    = Manhattan Voronoi Diagram.svg
 | alt2      = Voronoi diagram under Manhattan distance
 | caption2  = [[Manhattan distance]]
}}