In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" distance between two points that one would measure with a ruler, and is given by the Pythagorean formula. By using this formula as distance, Euclidean space (or even any inner product space) becomes a metric space. The associated norm is called the Euclidean norm. Older literature refers to the metric as Pythagorean metric.
The Euclidean distance between points p and q is the length of the line segment . In Cartesian coordinates, if p = (p1, p2,..., pn) and q = (q1, q2,..., qn) are two points in Euclidean n-space, then the distance from p to q is given by:
The Euclidean norm measures the distance of a point to the origin of Euclidean space:
where the last equation involves the dot product. This is the length of p, when regarded as a Euclidean vector from the origin. The distance itself is given by:
In one dimension, the distance between two points on the real line is the absolute value of their numerical difference. Thus if x and y are two points on the real line, then the distance between them is computed as
In one dimension, there is a single homogeneous, translation-invariant metric (in other words, a distance that is induced by a norm), up to a scale factor of length, which is the Euclidean distance. In higher dimensions there are other possible norms.
In the Euclidean plane, if p = (p1, p2) and q = (q1, q2) then the distance is given by
Alternatively, it follows from () that if the polar coordinates of the point p are (r1, θ1) and those of q are (r2, θ2), then the distance between the points is
In three-dimensional Euclidean space, the distance is
and so on.
- Mahalanobis distance normalizes based on a covariance matrix to make the distance metric scale-invariant.
- Manhattan distance measures distance following only axis-aligned directions.
- Chebyshev distance measures distance assuming only the most significant dimension is relevant.
- Minkowski distance is a generalization that unifies Euclidean distance, Manhattan distance, and Chebyshev distance.
- Pythagorean addition