Euclidean distance

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). 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.

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:

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

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

${\sqrt {(p_{x}-q_{x})^{2}+(p_{y}-q_{y})^{2}+(p_{z}-q_{z})^{2}}}.$

See also

• Mahalanobis distance
• Manhattan distance
• Metric