Chebyshev distance
High school
Consider two points $p = (p_1, \dots, p_n)$ and $q = (q_1, \dots, q_n)$ in $\reals^n$. The Chebyshev distance between $p$ and $q$ is the largest difference in any one coordinate:
\[d_{\infty}(p, q) = \max_i \abs{p_i - q_i}.\]Examples
Worked examples
- The north pole $p = (0, 0, 1)$ and the point $q = (1, 0, 0)$ on the equator of the unit sphere in $\reals^3$ : $d_{\infty}(p, q) = \max(1, 0, 1) = 1$ High school
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The points $p = (0, 0)$ and $q = (100, 100)$ in $\reals^2$
: $d_{\infty}(p, q) = \max(100, 100) = 100$
High school
The points $(0, 0)$ and $(100, 100)$ are far apart in the Euclidean, taxicab, and Chebyshev distances, which differ by factors of up to $2$.
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The points $p = (0, 0)$ and $q = (2, 0)$ in $\reals^2$
: $d_{\infty}(p, q) = \max(2, 0) = 2$
High school
The points $(0, 0)$ and $(2, 0)$ lie on a line parallel to an axis, so the Euclidean, taxicab, and Chebyshev distances all equal $2$.
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The points $p = (1, 1)$ and $q = (4, 5)$ in $\reals^2$
: $d_{\infty}(p, q) = \max(3, 4) = 4$
High school
The points $(1, 1)$ and $(4, 5)$ differ by $3$ in the first coordinate and $4$ in the second.
Properties
- The Chebyshev distance is $\norm{p - q}_{\infty}$, the infinity norm of the difference.
- On a chessboard, the Chebyshev distance between two squares is the number of moves a king needs to go from one to the other.
Distinctions involving this term
- Types of distances: A metric measures how far apart two points are, and the same two points can be at different distances under different metrics. The Euclidean distance is the straight-line distance, the taxicab distance travels along the axes, and the Chebyshev distance takes the largest difference in one coordinate. The geodesic distance is the length of the shortest path that stays on a surface, such as the great-circle distance on the Earth. The discrete metric only says whether two points are equal.