# isometry

For the mechanical engineering and architecture usage, see isometric projection. For isometry in differential geometry, see isometry (Riemannian geometry).

In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces. Geometric figures which can be related by an isometry are called congruent.

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.

## Definitions

The notion of isometry comes in two main flavors: global isometry and a weaker notion path isometry or arcwise isometry. Both are often called just isometry and one should guess from context which one is intended.

Let and be metric spaces with metrics and . A map is called distance preserving if for any one has A distance preserving map is automatically injective.

A global isometry is a bijective distance preserving map. A path isometry or arcwise isometry is a map which preserves the lengths of curves (not necessarily bijective).

Two metric spaces X and Y are called isometric if there is an isometry from X to Y. The set of isometries from a metric space to itself forms a group with respect to function composition, called the isometry group.

## Linear isometries

Given two normed vector spaces V and W, a linear isometry is a linear map f : VW that preserves the norms:
for all v in V. Linear isometries are distance-preserving maps in the above sense. They are global isometries if and only if they are surjective.

## Generalizations

• Given a positive real number ε, an ε-isometry or almost isometry (also called a Hausdorff approximation) is a map between metric spaces such that
• for one has , and
• for any point there exists a point with
That is, an ε-isometry preserves distances to within ε and leaves no element of the codomain further than ε away from the image of an element of the domain. Note that ε-isometries are not assumed to be continuous.

Mechanical Engineering is an engineering discipline that involves the application of principles of physics for analysis, design, manufacturing, and maintenance of mechanical systems.
Architecture is the art and science of designing buildings and structures. A wider definition often includes the design of the total built environment: from the macrolevel of town planning, urban design, and landscape architecture to the microlevel of construction details and,
Isometric projection is a form of graphical projection — more specifically, an axonometric projection. It is a method of visually representing three-dimensional objects in two dimensions, in which the three coordinate axes appear equally foreshortened and the angles between
In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. It arises naturally from the study of the theory of differential equations. Differential geometry is the study of geometry using differential calculus (cf.
In the study of Riemannian geometry in mathematics, a local isometry from one (pseudo-)Riemannian manifold to another is a map which pulls back the metric tensor on the second manifold to the metric tensor on the first.
Mathematics (colloquially, maths or math) is the body of knowledge centered on such concepts as quantity, structure, space, and change, and also the academic discipline that studies them. Benjamin Peirce called it "the science that draws necessary conclusions".
Distance is a numerical description of how far apart objects are at any given moment in time. In physics or everyday discussion, distance may refer to a physical length, a period of time, or an estimation based on other criteria (e.g. "two counties over").
In mathematics, an isomorphism (Greek: isos "equal", and morphe "shape") is a bijective map f such that both f and its inverse f −1 are homomorphisms, i.e., structure-preserving mappings.
In mathematics, a metric space is a set where a notion of distance (called a metric) between elements of the set is defined.

The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space.
congruent if one can be transformed into the other by an isometry, i.e., a combination of translations, rotations and reflections. In less formal language, two sets are congruent if they have the same shape and size, but are in different positions (for instance one may be rotated,
embedding (or imbedding) is one instance of some mathematical object contained within another instance, such as a group that is a subgroup.

## Abstractly or categorically

An abstract embedding
equivalence class of an element a in X is the subset of all elements in X which are equivalent to a:
[a] =

The notion of equivalence classes is useful for constructing sets out of already constructed ones.
Cauchy sequence, named after Augustin Cauchy, is a sequence whose elements become close to each other as the sequence progresses. To be more precise, by dropping enough (but still only a finite number of) terms from the start of the sequence, it is possible to make the
For Cauchy completion in category theory, see Karoubi envelope.

In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in
In topology and related branches of mathematics, a closed set is a set whose complement is open.

## Definition of a closed set

In a metric space, a set is closed if every limit point of the set is a point in the set.
Rn. It turns out that the following properties of "vector length" are the crucial ones.
1. The zero vector, 0, has zero length; every other vector has a positive length.

In mathematics, Banach spaces (pronounced ['banaɣ]), named after Stefan Banach, are one of the central objects of study in functional analysis.
In mathematics, a metric space is a set where a notion of distance (called a metric) between elements of the set is defined.

The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space.
function expresses dependence between two quantities, one of which is given (the independent variable, argument of the function, or its "input") and the other produced (the dependent variable, value of the function, or "output").
non-injective function.]] In mathematics, an injective function is a function which associates distinct arguments to distinct values. More precisely, a function f is said to be injective if it maps distinct x in the domain to distinct y
In mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property that, for every y in Y, there is exactly one x in X such that
f(x) = y.
In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. A simple example is the circle.
SET may stand for:
• Sanlih Entertainment Television, a television channel in Taiwan
• Secure electronic transaction, a protocol used for credit card processing,

group is a set with a binary operation that satisfies certain axioms, detailed below. For example, the set of integers with addition is a group. The branch of mathematics which studies groups is called group theory.
composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite.