Euclidean distance

n = 3, formula results from repeated application of the Pythagorean theorem

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. ${\displaystyle n}$

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: ${\displaystyle d(p,q)}$ ${\displaystyle n}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \|q-p\|_{2}}$ ${\displaystyle p}$ ${\displaystyle q}$ ${\displaystyle p=(p_{1},\ldots ,p_{n})}$ ${\displaystyle q=(q_{1},\ldots ,q_{n})}$

${\displaystyle d(p,q)=\|q-p\|_{2}={\sqrt {(q_{1}-p_{1})^{2}+\cdots +(q_{n}-p_{n})^{2}}}={\sqrt {\sum _{i=1}^{n}(q_{i}-p_{i})^{2}}}}$

n = 2, corresponds to the Pythagorean theorem

A well-known special case of calculating a Euclidean distance for is the Pythagorean theorem . ${\displaystyle n=2}$

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 ${\displaystyle p=(2,3,-1)}$ ${\displaystyle q=(4,1,-2)}$

${\displaystyle d(p,q)={\sqrt {(4-2)^{2}+(1-3)^{2}+(-2-(-1))^{2}}}={\sqrt {2^{2}+(-2)^{2}+(-1)^{2}}}={\sqrt {9}}=3}$.

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).