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.

Title Euclidean distance
Canonical name EuclideanDistance
Date of creation 2013-03-22 12:08:21
Last modified on 2013-03-22 12:08:21
Owner drini (3)
Last modified by drini (3)
Numerical id 15
Author drini (3)
Entry type Definition
Classification msc 53A99
Classification msc 54E35
Synonym Euclidean metric
Synonym standard metric
Synonym standard topology
Synonym Euclidean
Synonym canonical topology
Synonym usual topology
Related topic Topology
Related topic BoundedInterval
Related topic EuclideanVectorSpace
Related topic DistanceOfNonParallelLines
Related topic EuclideanVectorSpace2
Related topic Hyperbola2
Related topic CassiniOval