Chebyshev distance

In mathematics, Chebyshev distance (or Tchebychev distance), maximum metric, or L_∞ metric is a metric defined on a real coordinate space where the distance between two points is the greatest of their differences along any coordinate dimension. It is named after Pafnuty Chebyshev.

It is also known as chessboard distance, since in the game of chess the minimum number of moves needed by a king to go from one square on a chessboard to another equals the Chebyshev distance between the centers of the squares, if the squares have side length one, as represented in 2-D spatial coordinates with axes aligned to the edges of the board. For example, the Chebyshev distance between f6 and e2 equals 4.

The Chebyshev distance between two vectors or points a and b, with standard coordinates ${\displaystyle a_{i}}$ and ${\displaystyle b_{i}}$, respectively, is

$${\displaystyle D(a,b)=\max _{i}(|a_{i}-b_{i}|).}$$ This equals the limit of the L_p metrics: $${\displaystyle D(a,b)=\lim _{p\to \infty }{\bigg (}\sum _{i=1}^{n}\left|a_{i}-b_{i}\right|^{p}{\bigg )}^{1/p},}$$ hence it is also known as the L_∞ metric.

Mathematically, the Chebyshev distance is a metric induced by the supremum norm or uniform norm. It is an example of an injective metric.

In two dimensions, i.e. plane geometry, if the points a and b have Cartesian coordinates ${\displaystyle (x_{1},y_{1})}$ and ${\displaystyle (x_{2},y_{2})}$, their Chebyshev distance is

$${\displaystyle D_{\rm {Chebyshev}}=\max \left(\left|x_{2}-x_{1}\right|,\left|y_{2}-y_{1}\right|\right).}$$

Under this metric, a circle of radius r, which is the set of points with Chebyshev distance r from a center point, is a square whose sides have the length 2r and are parallel to the coordinate axes.