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Metric

A nonnegative function g(x,y) describing the "distance" between neighboring points for a given set. A metric satisfies the triangle inequality

g(x,y)+g(y,z)>=g(x,z)
(1)

and is symmetric, so

g(x,y)=g(y,x).
(2)

A metric also satisfies

g(x,x)=0,
(3)

as well as the condition that g(x,y)=0 implies x=y. If this latter condition is dropped, then g(x,y) is called a pseudometric instead of a metric.

A set possessing a metric is called a metric space. When viewed as a tensor, the metric is called a metric tensor.

See also

Cayley-Klein-Hilbert Metric, Distance, Equivalent Metrics, French Metro Metric, Fundamental Forms, Hedgehog Metric, Hyperbolic Metric, Metric Entropy, Metric Equivalence Problem, Metric Space, Metric Tensor, Metric Topology, Part Metric, Pseudometric, Riemannian Metric, Taxicab Metric, Ultrametric Explore this topic in the MathWorld classroom

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References

Gray, A. "Metrics on Surfaces." Ch. 15 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 341-358, 1997.

Referenced on Wolfram|Alpha

Metric

Cite this as:

Weisstein, Eric W. "Metric." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Metric.html

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