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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, and is given by the [[Pythagorean theorem|Pythagorean formula]]. By using this formula as distance, Euclidean space (or even any [[inner product space]]) becomes a [[metric space]]. The associated [[Norm (mathematics)|norm]] is called the '''[[Norm (mathematics)#Euclidean norm|Euclidean norm]].''' Older literature refers to the metric as '''Pythagorean metric'''.
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| ==Definition==
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| The '''Euclidean distance''' between points '''p''' and '''q''' is the length of the [[line segment]] connecting them (<math>\overline{\mathbf{p}\mathbf{q}}</math>).
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| In [[Cartesian coordinates]], if '''p''' = (''p''<sub>1</sub>, ''p''<sub>2</sub>,..., ''p''<sub>''n''</sub>) and '''q''' = (''q''<sub>1</sub>, ''q''<sub>2</sub>,..., ''q''<sub>''n''</sub>) are two points in [[Euclidean space|Euclidean ''n''-space]], then the distance from '''p''' to '''q''', or from '''q''' to '''p''' is given by:
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| {{NumBlk|:|<math>\mathrm{d}(\mathbf{p},\mathbf{q}) = \mathrm{d}(\mathbf{q},\mathbf{p}) = \sqrt{(q_1-p_1)^2 + (q_2-p_2)^2 + \cdots + (q_n-p_n)^2} = \sqrt{\sum_{i=1}^n (q_i-p_i)^2}.</math>|{{EquationRef|1}}}}
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| The position of a point in a Euclidean ''n''-space is a [[Euclidean vector]]. So, '''p''' and '''q''' are Euclidean vectors, starting from the origin of the space, and their tips indicate two points. The '''[[Euclidean norm]]''', or '''Euclidean length''', or '''magnitude''' of a vector measures the length of the vector:
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| :<math>\|\mathbf{p}\| = \sqrt{p_1^2+p_2^2+\cdots +p_n^2} = \sqrt{\mathbf{p}\cdot\mathbf{p}}</math>
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| where the last equation involves the [[dot product]].
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| A vector can be described as a directed line segment from the [[Origin (mathematics)|origin]] of the Euclidean space (vector tail), to a point in that space (vector tip). If we consider that its length is actually the distance from its tail to its tip, it becomes clear that the Euclidean norm of a vector is just a special case of Euclidean distance: the Euclidean distance between its tail and its tip.
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| The distance between points '''p''' and '''q''' may have a direction (e.g. from '''p''' to '''q'''), so it may be represented by another vector, given by
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| :<math>\mathbf{q} - \mathbf{p} = (q_1-p_1, q_2-p_2, \cdots, q_n-p_n)</math>
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| In a three-dimensional space (''n''=3), this is an arrow from '''p''' to '''q''', which can be also regarded as the position of '''q''' relative to '''p'''. It may be also called a [[displacement (vector)|displacement]] vector if '''p''' and '''q''' represent two positions of the same point at two successive instants of time.
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| The Euclidean distance between '''p''' and '''q''' is just the Euclidean length of this distance (or displacement) vector:
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| {{NumBlk|:|<math>\|\mathbf{q} - \mathbf{p}\| = \sqrt{(\mathbf{q}-\mathbf{p})\cdot(\mathbf{q}-\mathbf{p})}.</math>|{{EquationRef|2}}}}
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| which is equivalent to equation 1, and also to:
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| :<math>\|\mathbf{q} - \mathbf{p}\| = \sqrt{\|\mathbf{p}\|^2 + \|\mathbf{q}\|^2 - 2\mathbf{p}\cdot\mathbf{q}}.</math>
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| ===One dimension===
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| 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 given by:
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| :<math>\sqrt{(x-y)^2} = |x-y|.</math>
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| In one dimension, there is a single homogeneous, translation-invariant [[Metric (mathematics)|metric]] (in other words, a distance that is induced by a [[Norm (mathematics)|norm]]), up to a scale factor of length, which is the Euclidean distance. In higher dimensions there are other possible norms.
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| ===Two dimensions===
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| In the [[Euclidean plane]], if '''p''' = (''p''<sub>1</sub>, ''p''<sub>2</sub>) and '''q''' = (''q''<sub>1</sub>, ''q''<sub>2</sub>) then the distance is given by
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| :<math>\mathrm{d}(\mathbf{p},\mathbf{q})=\sqrt{(p_1-q_1)^2 + (p_2-q_2)^2}.</math> | |
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| This is equivalent to the [[Pythagorean theorem]].
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| Alternatively, it follows from ({{EquationRef|2}}) that if the [[polar coordinates]] of the point '''p''' are (''r''<sub>1</sub>, θ<sub>1</sub>) and those of '''q''' are (''r''<sub>2</sub>, θ<sub>2</sub>), then the distance between the points is
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| :<math>\sqrt{r_1^2 + r_2^2 - 2 r_1 r_2 \cos(\theta_1 - \theta_2)}.</math>
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| ===Three dimensions===
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| In three-dimensional Euclidean space, the distance is
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| :<math>d(p, q) = \sqrt{(p_1 - q_1)^2 + (p_2 - q_2)^2+(p_3 - q_3)^2}.</math>
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| ===N dimensions===
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| In general, for an n-dimensional space, the distance is
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| :<math>d(p, q) = \sqrt{(p_1- q_1)^2 + (p_2 - q_2)^2+...+(p_i - q_i)^2+...+(p_n - q_n)^2}.</math>
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| ===Squared Euclidean distance===
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| The standard Euclidean distance can be squared in order to place progressively greater weight on objects that are farther apart. In this case, the equation becomes
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| :<math>d^2(p, q) = (p_1 - q_1)^2 + (p_2 - q_2)^2+...+(p_i - q_i)^2+...+(p_n - q_n)^2.</math>
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| Squared Euclidean Distance is not a metric as it does not satisfy the [[triangle inequality]], however it is frequently used in optimization problems in which distances only have to be compared.
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| It is also referred to as [[quadrance]] within the field of [[rational trigonometry]].
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| ==See also==
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| *[[Chebyshev distance]] measures distance assuming only the most significant dimension is relevant.
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| *[[Mahalanobis distance]] normalizes based on a covariance matrix to make the distance metric scale-invariant.
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| *[[Manhattan distance]] measures distance following only axis-aligned directions.
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| *[[Metric (mathematics)|Metric]]
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| *[[Minkowski distance]] is a generalization that unifies Euclidean distance, Manhattan distance, and Chebyshev distance.
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| *[[Pythagorean addition]]
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| ==References==
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| * Elena Deza & Michel Marie Deza (2009) ''Encyclopedia of Distances'', page 94, Springer.
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| * http://www.statsoft.com/textbook/cluster-analysis/, March 2, 2011
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| {{DEFAULTSORT:Euclidean Distance}}
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| [[Category:Metric geometry]]
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| [[Category:Length]]
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| [[Category:String similarity measures]]
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