Euclidean distance

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.

Definition

The Euclidean distance between points p and q is the length of the line segment connecting them (${\overline {\mathbf {p} \mathbf {q} }}$).

In Cartesian coordinates, if p = (p₁, p₂,..., p_n) and q = (q₁, q₂,..., q_n) are two points in Euclidean n-space, then the distance from p to q, or from q to p is given by:

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

The position of a point in a Euclidean n-space is an Euclidean vector. So, p and q are Euclidean vectors, starting from the origin of the space, and their tips indicate two points. The Euclidean norm, or Euclidean length, or magnitude of a vector measures the length of the vector:

$\|\mathbf {p} \|={\sqrt {p_{1}^{2}+p_{2}^{2}+\cdots +p_{n}^{2}}}={\sqrt {\mathbf {p} \cdot \mathbf {p} }}$

where the last equation involves the dot product.

A vector can be described as a directed line segment from the origin of the Euclidean space (vector tail), to a point in that space (vector tip). If we consider that its length is actually the distance from its tail to its tip, it becomes clear that the Euclidean norm of a vector is just a special case of Euclidean distance: the Euclidean distance between its tail and its tip.

The distance between points p and q may have a direction (e.g. from p to q), so it may be represented by another vector, given by

$\mathbf {q} -\mathbf {p} =(q_{1}-p_{1},q_{2}-p_{2},\cdots ,q_{n}-p_{n})$

In a three-dimensional space (n=3), this is an arrow from p to q, which can be also regarded as the position of q relative to p. It may be also called a displacement vector if p and q represent two positions of the same point at two successive instants of time.

The Euclidean distance between p and q is just the Euclidean length of this distance (or displacement) vector:

$\|\mathbf {q} -\mathbf {p} \|={\sqrt {(\mathbf {q} -\mathbf {p} )\cdot (\mathbf {q} -\mathbf {p} )}}.$ (2)

which is equivalent to equation 1, and also to:

$\|\mathbf {q} -\mathbf {p} \|={\sqrt {\|\mathbf {p} \|^{2}+\|\mathbf {q} \|^{2}-2\mathbf {p} \cdot \mathbf {q} }}.$

One dimension

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

${\sqrt {(x-y)^{2}}}=|x-y|.$

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.

Two dimensions

In the Euclidean plane, if p = (p₁, p₂) and q = (q₁, q₂) then the distance is given by

$\mathrm {d} (\mathbf {p} ,\mathbf {q} )={\sqrt {(p_{1}-q_{1})^{2}+(p_{2}-q_{2})^{2}}}.$

Alternatively, it follows from (2) 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

${\sqrt {r_{1}^{2}+r_{2}^{2}-2r_{1}r_{2}\cos(\theta _{1}-\theta _{2})}}.$

Three dimensions

In three-dimensional Euclidean space, the distance is

$d(x,y)={\sqrt {(x_{1}-y_{1})^{2}+(x_{2}-y_{2})^{2}+(x_{3}-y_{3})^{2}}}.$

See also

• 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.
• Metric
• Pythagorean addition

References

• Elena Deza & Michel Marie Deza (2009) Encyclopedia of Distances, page 94, Springer.