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In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" (i.e. straight-line) distance between two points in Euclidean space. With this distance, Euclidean space becomes a metric space. The associated norm is called the Euclidean norm. Older literature refers to the metric as Pythagorean metric. A generalized term for the Euclidean norm is the L2 norm or L2 distance. ==Definition== The Euclidean distance between points p and q is the length of the line segment connecting them (). In Cartesian coordinates, if p = (''p''1, ''p''2,..., ''p''''n'') and q = (''q''1, ''q''2,..., ''q''''n'') are two points in Euclidean ''n''-space, then the distance (d) from p to q, or from q to p is given by the Pythagorean formula: 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: : where the last equation involves the dot product. A vector can be described as a directed line segment from the 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. 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 : 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 if p and q represent two positions of the same point at two successive instants of time. The Euclidean distance between p and q is just the Euclidean length of this distance (or displacement) vector: which is equivalent to equation 1, and also to: : 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Euclidean distance」の詳細全文を読む スポンサード リンク
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