# Chebyshev distance

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In mathematics, **Chebyshev distance** (or **Tchebychev distance**), **maximum metric**, or L_{∞} metric^{[1]} is a metric defined on a vector space where the distance between two vectors is the greatest of their differences along any coordinate dimension.^{[2]} 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.^{[3]} For example, the Chebyshev distance between f6 and e2 equals 4.

## Definition

The Chebyshev distance between two vectors or points *p* and *q*, with standard coordinates and , respectively, is

This equals the limit of the L_{p} metrics:

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 *p* and *q* have Cartesian coordinates
and , their Chebyshev distance is

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 2*r* and are parallel to the coordinate axes.

On a chess board, where one is using a *discrete* Chebyshev distance, rather than a continuous one, the circle of radius *r* is a square of side lengths 2*r,* measuring from the centers of squares, and thus each side contains 2*r*+1 squares; for example, the circle of radius 1 on a chess board is a 3×3 square.

## Properties

In one dimension, all L_{p} metrics are equal – they are just the absolute value of the difference.

The two dimensional Manhattan distance also has circles in the form of squares, with sides of length √*2**r*, oriented at an angle of π/4 (45°) to the coordinate axes, so the planar Chebyshev distance can be viewed as equivalent by rotation and scaling to the planar Manhattan distance.

However, this equivalence between L_{1} and L_{∞} metrics does not generalize to higher dimensions. A sphere formed using the Chebyshev distance as a metric is a cube with each face perpendicular to one of the coordinate axes, but a sphere formed using Manhattan distance is an octahedron: these are dual polyhedra, but among cubes, only the square (and 1-dimensional line segment) are self-dual polytopes.

The Chebyshev distance is sometimes used in warehouse logistics,^{[4]} as it effectively measures the time an overhead crane takes to move an object (as the crane can move on the x and y axes at the same time but at the same speed along each axis).

On a grid (such as a chessboard), the points at a Chebyshev distance of 1 of a point are the Moore neighborhood of that point.

## See also

## References

- ↑ Cyrus. D. Cantrell (2000).
*Modern Mathematical Methods for Physicists and Engineers*. Cambridge University Press. ISBN 0-521-59827-3. - ↑ James M. Abello, Panos M. Pardalos, and Mauricio G. C. Resende (editors) (2002).
*Handbook of Massive Data Sets*. Springer. ISBN 1-4020-0489-3. - ↑ David M. J. Tax; Robert Duin; Dick De Ridder (2004).
*Classification, Parameter Estimation and State Estimation: An Engineering Approach Using MATLAB*. John Wiley and Sons. ISBN 0-470-09013-8. - ↑ André Langevin; Diane Riopel (2005).
*Logistics Systems*. Springer. ISBN 0-387-24971-0.