Euclidean distance

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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.

1 Definition
2 One-dimensional distance
3 Two-dimensional distance
4 Three-dimensional distance
5 See also

[edit] 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:

$\sqrt{(p_1-q_1)^2 + (p_2-q_2)^2 + \cdots + (p_n-q_n)^2} = \sqrt{\sum_{i=1}^n (p_i-q_i)^2}.$

[edit] 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.

[edit] 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: