A nonnegative function
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)](/images/equations/Metric/NumberedEquation1.svg) |
(1)
|
and is symmetric, so
![g(x,y)=g(y,x).](/images/equations/Metric/NumberedEquation2.svg) |
(2)
|
A metric also satisfies
![g(x,x)=0,](/images/equations/Metric/NumberedEquation3.svg) |
(3)
|
as well as the condition that
implies
. If this latter condition is dropped,
then
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
Explore with Wolfram|Alpha
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
Subject classifications