In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" distance between two points that one would measure with a ruler, which can be proven by repeated application of the Pythagorean theorem. By using this formula as distance, Euclidean space becomes a metric space (even a Hilbert space). The associated norm is called the Euclidean norm.

Older literature refers to this metric as Pythagorean metric. The technique has been rediscovered numerous times throughout history, as it is a logical extension of the Pythagorean theorem.

Definition

The Euclidean distance between points $P=(p\_1,p\_2,dots,p\_n),$ and $Q=(q\_1,q\_2,dots,q\_n),$, in Euclidean n-space, is defined as:

$sqrt\{(p\_1-q\_1)^2\; +\; (p\_2-q\_2)^2\; +\; cdots\; +\; (p\_n-q\_n)^2\}\; =\; sqrt\{sum\_\{i=1\}^n\; (p\_i-q\_i)^2\}.$

One-dimensional distance

For two 1D points, $P=(p\_x),$ and $Q=(q\_x),$, the distance is computed as:

$sqrt\{(p\_x-q\_x)^2\}\; =\; |\; p\_x-q\_x\; |$

The absolute value signs are used since distance is normally considered to be an unsigned scalar value.

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-dimensional distance

For two 2D points, $P=(p\_x,p\_y),$ and $Q=(q\_x,q\_y),$, the distance is computed as:

$sqrt\{(p\_x-q\_x)^2\; +\; (p\_y-q\_y)^2\}$

Alternatively, expressed in circular coordinates (also known as polar coordinates), using $P=(r\_1,\; theta\_1),$ and $Q=(r\_2,\; theta\_2),$, the distance can be computed as:

$sqrt\{r\_1^2\; +\; r\_2^2\; -\; 2\; r\_1\; r\_2\; cos(theta\_1\; -\; theta\_2)\}$

Three-dimensional distance

For two 3D points, $P=(p\_x,p\_y,p\_z),$ and $Q=(q\_x,q\_y,q\_z),$, the distance is computed as

$sqrt\{(p\_x-q\_x)^2\; +\; (p\_y-q\_y)^2+(p\_z-q\_z)^2\}.$

N-dimensional distance

For two N-D points, $P=(p\_1,p\_2,...,p\_n),$ and $Q=(q\_1,q\_2,...,q\_n),$, the distance is computed as

$sqrt\{(p\_1-q\_1)^2\; +\; (p\_2-q\_2)^2+...+(p\_n-q\_n)^2\}.$

See also

• Mahalanobis distance
• Manhattan distance
• Metric
• Pythagorean addition