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isometry
In mathematics, an isometry is a distance-preserving injective map between metric spaces.
==Introduction==
Given a metric space (loosely, a set and a scheme for assigning distances between elements of the set), an isometry is a transformation which maps elements to the same or another metric space such that the distance between the image elements in the new metric space is equal to the distance between the elements in the original metric space. In a two-dimensional or three-dimensional Euclidean space, two geometric figures are congruent if they are related by an isometry: related by either a rigid motion (translation or rotation), or a composition of a rigid motion and a reflection. They are equal, up to an action of a rigid motion, if related by a direct isometry (orientation preserving).
Isometries are often used in constructions where one space is embedded in another space. For instance, the completion of a metric space ''M'' involves an isometry from ''M'' into ''M, a quotient set of the space of Cauchy sequences on ''M''. The original space ''M'' is thus isometrically isomorphic to a subspace of a complete metric space, and it is usually identified with this subspace. Other embedding constructions show that every metric space is isometrically isomorphic to a closed subset of some normed vector space and that every complete metric space is isometrically isomorphic to a closed subset of some Banach space.
An isometric surjective linear operator on a Hilbert space is called a unitary operator.
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「isometry」の詳細全文を読む
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