1. Definition

2. Properties

3. See also

4. References

5. External links

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 2r 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 2r, measuring from the centers of squares, and thus each side contains 2r+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 {{sqrt|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 L₁ 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

• King's graph
• Uniform norm

References

External links

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