isometry

Information about 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 $\text{[math]}$ and $\text{[math]}$ be metric spaces with metrics $\text{[math]}$ and $\text{[math]}$ . A map $\text{[math]}$ is called distance preserving if for any $\text{[math]}$ one has $\text{[math]}$ 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.

Examples

Any reflection, translation and rotation is a global isometry on Euclidean spaces. See also Euclidean group.
The map R $\text{[math]}$ R defined by $\text{[math]}$ is a path isometry but not a global isometry.
The isometric linear maps from Cⁿ to itself are the unitary matrices.

Linear isometries

Given two normed vector spaces V and W, a linear isometry is a linear map f : V → W that preserves the norms:

$\text{[math]}$

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 $\text{[math]}$ between metric spaces such that
for $\text{[math]}$ one has $\text{[math]}$ , and
for any point $\text{[math]}$ there exists a point $\text{[math]}$ with $\text{[math]}$

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.

Quasi-isometry is yet another useful generalization.

Abstractly or categorically

An abstract embedding
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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.
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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
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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
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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.
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Rⁿ. It turns out that the following properties of "vector length" are the crucial ones.

The zero vector, 0, has zero length; every other vector has a positive length.

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In mathematics, Banach spaces (pronounced ['banaɣ]), named after Stefan Banach, are one of the central objects of study in functional analysis.
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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").
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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
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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.
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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.
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SET may stand for:

Sanlih Entertainment Television, a television channel in Taiwan
Secure electronic transaction, a protocol used for credit card processing,

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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.
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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.
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In mathematics, the isometry group of a metric space is the set of all isometries from the metric space onto itself, with the function composition as group operation. Its identity element is the identity function.
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reflection (also spelled reflexion) is a map that transforms an object into its mirror image. For example, a reflection of the small English letter p in respect to a vertical line would look like q.
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translation is moving every point a constant distance in a specified direction. It is one of the rigid motions (other rigid motions include rotation and reflection). A translation can also be interpreted as the addition of a constant vector to every point, or as shifting the origin
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This article is about rotation as a movement of a physical body. For other uses, see Rotation (disambiguation).

A rotation is a movement of an object in a circular motion.
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In mathematics, a linear map (also called a linear transformation or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication.
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