Euclidean distance
The Euclidean distance is the concept of distance in Euclidean geometry . The Euclidean distance between two points in a plane or in space is the length of a line that connects these two points, measured with a ruler, for example . This distance is invariant under movements ( congruence maps ).
In Cartesian coordinates , the Euclidean distance can be calculated using the Pythagorean theorem . With the help of the formula obtained in this way, the concept of the Euclidean distance can be generalized to - dimensional Euclidean and unitary vector spaces , Euclidean point spaces and coordinate spaces.
This distance is called "Euclidean" to differentiate it from more general distance terms , such as:
- that of hyperbolic geometry ,
- that of Riemannian geometry ,
- Distances in normalized vector spaces ,
- Distances in any metric spaces .
Euclidean space
In the two-dimensional Euclidean plane or in three-dimensional Euclidean space with Euclidean distance agrees with the ideological distance match. In the more general case of - dimensional Euclidean space , it is defined for two points or vectors by the Euclidean norm of the difference vector between the two points. If the points and are given by the coordinates and , then:
A well-known special case of calculating a Euclidean distance for is the Pythagorean theorem .
The Euclidean distance is a metric and in particular it satisfies the triangle inequality . In addition to the Euclidean distance, there are a number of other distance measures. Since the Euclidean distance comes from a norm , namely the Euclidean norm, it is translation-invariant .
In statistics , the Euclidean distance is a special case of the weighted Euclidean distance and its square is a special case of the Mahalanobis distance .
example
The Euclidean distance between the two points and is
- .
literature
- Hermann Schichl , Roland Steinbauer: Introduction to mathematical work . 2nd revised edition. Springer, 2012, ISBN 978-3-642-28646-9 , p. 382 ff.
- Winfried Schröter: Newer statistical procedures and modeling in geoecology . Springer, 2013, ISBN 978-3-322-83735-6 , pp. 120 ff.
- Elena Deza, Michel Marie Deza: Encyclopedia of Distances . Springer, 2009, ISBN 978-3-642-00233-5 , p. 94
Web links
- Eric W. Weisstein : Distance . In: MathWorld (English).
- Eric W. Weisstein : Euclidean Metric . In: MathWorld (English).