Isometry - WikiHQ

thumb|upright=1.4|A [[Function composition|composition of two opposite isometries is a direct isometry. A reflection in a line is an opposite isometry, like (reflection w.r.t the center diagonal line) or (reflection w.r.t the right diagonal line) on the image. Translation is a direct isometry: a rigid motion.]]

In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος isos meaning "equal", and μέτρον metron meaning "measure". If the transformation is from a metric space to itself, it is a kind of geometric transformation known as a motion.

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;

the isometry that relates them is either a rigid motion (translation or rotation), or a composition of a rigid motion and a reflection.

Isometries are often used in constructions where one space is embedded in another space. For instance, the completion of a metric space <math>M </math> involves an isometry from <math>M </math> into <math>M',</math> a quotient set of the space of Cauchy sequences on <math>M.</math>

The original space <math>M </math> 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.

Definition

Let <math>X</math> and <math>Y</math> be metric spaces with metrics (e.g., distances) <math display="inline">d_X </math> and <math display="inline">d_Y.</math> A map <math display="inline">f\colon X \to Y </math> is called an isometry or distance-preserving map if for any <math>a, b \in X</math>,

:<math>d_X(a,b)=d_Y\!\left(f(a),f(b)\right).</math>

Bibliography