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 $\overline{\mathbf{p}\mathbf{q}}$. 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 is given by:

$\mathrm{d}(\mathbf{p},\mathbf{q}) = \sqrt{(p_1-q_1)^2 + (p_2-q_2)^2 + \cdots + (p_n-q_n)^2} = \sqrt{\sum_{i=1}^n (p_i-q_i)^2}$

The Euclidean norm measures the distance of a point to the origin of Euclidean space:

$\|\mathbf{p}\| = \sqrt{p_1^2+p_2^2+\cdots +p_n^2} = \sqrt{\mathbf{p}\cdot\mathbf{p}}$

where the last equation involves the dot product. This is the length of p, when regarded as a Euclidean vector from the origin. The distance itself is given by:

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

Special cases

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.

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

$d(\mathbf{p},\mathbf{q})=\sqrt{(p_1-q_1)^2 + (p_2-q_2)^2}.$

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

$\sqrt{r_1^2 + r_2^2 - 2 r_1 r_2 \cos(\theta_1 - \theta_2)}.$

In three-dimensional Euclidean space, the distance is

$\sqrt{(p_1-q_1)^2 + (p_2-q_2)^2+(p_3-q_3)^2}$

and so on.

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