Euclidean distance

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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.

This distance is called "Euclidean" to differentiate it from more general distance terms , such as:

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:

n = 2, corresponds to the Pythagorean theorem

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 .


The Euclidean distance between the two points and is



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