partial Euclidean distance (PED)Inmathematics, theEuclidean distanceorEuclidean metricis the "ordinary" (i.e straight line)distancebetween two points inEuclidean space. With this distance, Euclidean space becomes ametric space. The associatednormis called theEuclidean norm.Older literature refers to the metric asPythagorean metric.Contents 1Definition 1.1One dimension 1.2Two dimensions 1.3Three dimensions 1.4ndimensions 1.5Squared Euclidean distance 2See also 3ReferencesDefinition[edit]TheEuclidean distancebetween pointspandqis the length of theline segmentconnecting them ().InCartesian coordinates, ifp=(p1,p2,...,pn) andq=(q1,q2,...,qn) are two points inEuclideann-space, then the distance (d) fromptoq, or fromqtopis given by thePythagorean formula:

(1)

The position of a point in a Euclideann-space is aEuclidean vector. So,pandqare Euclidean vectors, starting from the origin of the space, and their tips indicate two points. TheEuclidean norm, orEuclidean length, ormagnitudeof a vector measures the length of the vector:

where the last equation involves thedot product.A vector can be described as a directed line segment from theoriginof 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.The distance between pointspandqmay have a direction (e.g. fromptoq), so it may be represented by another vector, given by

In a three-dimensional space (n=3), this is an arrow fromptoq, which can be also regarded as the position ofqrelative top. It may be also called adisplacementvector ifpandqrepresent two positions of the same point at two successive instants of time.The Euclidean distance betweenpandqis just the Euclidean length of this distance (or displacement) vector:

(2)

which is equivalent to equation 1, and also to:

One dimension[edit]In one dimension, the distance between two points on thereal lineis theabsolute valueof their numerical difference. Thus ifxandyare two points on the real line, then the distance between them is given by:

In one dimension, there is a single homogeneous, translation-invariantmetric(in other words, a distance that is induced by anorm), up to a scale factor of length, which is the Euclidean distance. In higher dimensions there are other possible norms.Two dimensions[edit]In theEuclidean plane, ifp=(p1,p2) andq=(q1,q2) then the distance is given by

This is equivalent to thePythagorean theorem.Alternatively, it follows from (2) that if thepolar coordinatesof the pointpare (r1,1) and those ofqare (r2,2), then the distance between the points is

Three dimensions[edit]In three-dimensional Euclidean space, the distance is

ndimensions[edit]In general, for ann-dimensional space, the distance is

Squared Euclidean distance[edit]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

Squared Euclidean Distance is not a metric as it does not satisfy thetriangle inequality, however it is frequently used in optimization problems in which distances only have to be compared.It is also referred to asquadrancewithin the field ofrational trigonometry.See also[edit] Chebyshev distancemeasures distance assuming only the most significant dimension is relevant. Euclidean distance matrix Hamming distanceidentifies the difference bit by bit of two strings Mahalanobis distancenormalizes based on a covariance matrix to make the distance metric scale-invariant. Manhattan distancemeasures distance following only axis-aligned directions. Metric Minkowski distanceis a generalization that unifies Euclidean distance, Manhattan distance, and Chebyshev distance. Pythagorean additionReferences[edit] Deza, Elena; Deza, Michel Marie (2009).Encyclopedia of Distances. Springer. p.94. "Cluster analysis". March 2, 2011.