- 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 thePythagorean theorem . By using this formula as distance, Euclidean space becomes ametric space (even aHilbert space ). The associated norm is called theEuclidean 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 $P=(r\_1,\; heta\_1),$ and $Q=(r\_2,\; heta\_2),$, the distance can be computed as::$sqrt\{r\_1^2\; +\; r\_2^2\; -\; 2\; r\_1\; r\_2\; cos(\; heta\_1\; -\; heta\_2)\}$

**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

*Wikimedia Foundation.
2010.*

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