Euclidean distance

The Euclidean distance is the distance concept of Euclidean geometry. The Euclidean distance between two points in the plane or in space is for example measured with a ruler, the length of a line that connects these two points. This distance is invariant under motions ( Kongruenzabbildungen ).

In Cartesian coordinates, the Euclidean distance using the Pythagorean theorem can be calculated. Using the formula obtained in this way, the concept of Euclidean distance in n-dimensional Euclidean and unitary vector spaces, Euclidean point spaces and coordinate spaces can be generalized.

" Euclidean " is this distance as opposed to more general distance terms, such as:

  • The hyperbolic geometry,
  • The Riemannian geometry,
  • Intervals in normed vector spaces,
  • Intervals in arbitrary metric spaces.

Euclidean space

In the two-dimensional Euclidean plane or in three-dimensional Euclidean space, the Euclidean distance with the ideological distance is the same. In the more general case of an n-dimensional Euclidean space defined for two points or vectors of the Euclidean norm of the difference vector between the two points. Are the points by the coordinates and, where, then:

A well-known special case of the calculation of a Euclidean distance for the theorem of Pythagoras.

The Euclidean distance is a metric and in particular satisfies the triangle inequality. In addition to the Euclidean distance, there are a number of other distance measures. Because the Euclidean distance from a norm arises, 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 is

  • Euclidean geometry
  • Multivariate Statistics