Euclidean distance
If and are two points on the plane, their Euclidean distance is given by
(1) |
Geometrically, it’s the length of the segment joining and , and also the norm of the difference vector (considering as vector space).
This distance induces a metric (and therefore a topology) on , called Euclidean metric (on ) or standard metric (on . The topology so induced is called standard topology or usual topology on and one basis can be obtained considering the set of all the open balls.
If and , then formula 1 can be generalized to by defining the Euclidean distance from to as
(2) |
Notice that this distance coincides with absolute value when . Euclidean distance on is also a metric (Euclidean or standard metric), and therefore we can give a topology, which is called the standard (canonical, usual, etc) topology of . The resulting (topological and vectorial) space is known as Euclidean space.
This can also be done for since as set and thus the metric on is the same given to , and in general, gets the same metric as .
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 |