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,..., pn) and q = (q1, q2,..., qn) 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, &theta;1) and those of q are (r2, &theta;2), 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.