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单词 EuclideanDistance
释义

Euclidean distance


If u=(x1,y1) and v=(x2,y2) are two points on the plane, their Euclidean distance is given by

(x1-x2)2+(y1-y2)2.(1)

Geometrically, it’s the length of the segment joining u and v, and also the norm of the difference vector (considering n as vector space).

This distancePlanetmathPlanetmath induces a metric (and therefore a topologyMathworldPlanetmath) on 2, called Euclidean metric (on R2) or standard metric (on R2). The topology so induced is called standard topology or usual topology on R2 and one basis can be obtained considering the set of all the open balls.

If a=(x1,x2,,xn) and b=(y1,y2,,yn), then formula 1 can be generalized to n by defining the Euclidean distance from a to b as

d(a,b)=(x1-y1)2+(x2-y2)2++(xn-yn)2.(2)

Notice that this distance coincides with absolute valueMathworldPlanetmathPlanetmathPlanetmathPlanetmath when n=1.Euclidean distance on n is also a metric (Euclidean or standard metric), and therefore we can give n a topology, which is called the standard (canonical, usual, etc) topology of n. The resulting (topological and vectorial) space is known as Euclidean space.

This can also be done for n since as set =2 and thus the metric on is the same given to 2, and in general, n gets the same metric as R2n.

TitleEuclidean distance
Canonical nameEuclideanDistance
Date of creation2013-03-22 12:08:21
Last modified on2013-03-22 12:08:21
Ownerdrini (3)
Last modified bydrini (3)
Numerical id15
Authordrini (3)
Entry typeDefinition
Classificationmsc 53A99
Classificationmsc 54E35
SynonymEuclidean metric
Synonymstandard metric
Synonymstandard topology
SynonymEuclidean
Synonymcanonical topology
Synonymusual topology
Related topicTopology
Related topicBoundedInterval
Related topicEuclideanVectorSpace
Related topicDistanceOfNonParallelLines
Related topicEuclideanVectorSpace2
Related topicHyperbola2
Related topicCassiniOval
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更新时间:2025/5/4 9:55:08