Isometry

For the mechanical engineering and architecture usage, see isometric projection. Mechanical Engineering is an Engineering discipline that involves the application of principles of physics for analysis Design, Manufacturing The term architecture (from Greek αρχιτεκτονικήarchitektoniki) can be used to mean a process a profession or documentation Isometric projection is a form of Graphical projection —more specifically an Axonometric projection. For isometry in differential geometry, see isometry (Riemannian geometry). Differential geometry is a mathematical discipline that uses the methods of differential and integral Calculus to study problems in Geometry In the study of Riemannian geometry in Mathematics, a local isometry from one ( pseudo - Riemannian manifold to another is a map which pulls

In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Distance is a numerical description of how far apart objects are In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined Geometric figures which can be related by an isometry are called congruent. In Geometry, two sets of points are called congruent if one can be transformed into the other by an Isometry, i

Isometries are often used in constructions where one space is embedded in another space. In Mathematics, an embedding (or imbedding) is one instance of some Mathematical structure contained within another instance such as a group 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. In Mathematical analysis, a Metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has In Mathematics, given a set X and an Equivalence relation ~ on X, the equivalence class of an element a in X In Mathematics, a Cauchy sequence, named after Augustin Cauchy, is a Sequence whose elements become arbitrarily close to each other as the sequence The original space M is thus isometrically isomorphic to a subspace of a complete metric space, and it is usually identified with this subspace. In Mathematical analysis, a Metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has 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. In Topology and related branches of Mathematics, a closed set is a set whose complement is open. In Mathematics, with 2- or 3-dimensional vectors with real -valued entries the idea of the "length" of a vector is intuitive and can easily be extended to In Mathematics, Banach spaces (ˈbanax named after Polish Mathematician Stefan Banach) are one of the central objects of study in Functional analysis

1 Definitions
2 Examples
3 Linear isometries
4 Generalizations
5 See also

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 determine from context which one is intended.

Let $X$ and $Y$ be metric spaces with metrics $d X$ and $d Y$ . In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined A map $f\colon X\to Y$ is called distance preserving if for any $x,y\in X$ one has $d_Y\left(f(x),f(y)\right)=d_X(x,y)$ . The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function A distance preserving map is automatically injective.

A global isometry is a bijective distance preserving map. In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property A path isometry or arcwise isometry is a map which preserves the lengths of curves (not necessarily bijective). In Mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object

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. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, a composite function represents the application of one function to the results of another 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

Examples

Any reflection, translation and rotation is a global isometry on Euclidean spaces. In Mathematics, a reflection (also spelled reflexion) is a map that transforms an object into its Mirror image. In Euclidean geometry, a translation is moving every point a constant distance in a specified direction A rotation is a movement of an object in a circular motion A two- Dimensional object rotates around a center (or point) of rotation See also Euclidean group. In Mathematics, the Euclidean group E ( n) sometimes called ISO( n) or similar is the Symmetry group of n -dimensional

The map R $\to$ R defined by $x\mapsto |x|$ is a path isometry but not a global isometry.

The isometric linear maps from Cⁿ to itself are the unitary matrices. In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that In Mathematics, a unitary matrix is an n by n complex matrix U satisfying the condition U^* U = UU^*

Linear isometries

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

$\|f(v)\| = \|v\|$

for all v in V. In Mathematics, with 2- or 3-dimensional vectors with real -valued entries the idea of the "length" of a vector is intuitive and can easily be extended to In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that Linear isometries are distance-preserving maps in the above sense. They are global isometries if and only if they are surjective. In Mathematics, a function f is said to be surjective or onto, if its values span its whole Codomain; that is for every

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
1. for $x,x'\in X$ one has $| d Y (f (x), f (x')) - d X (x, x') | < ε$ , and
2. for any point $y\in Y$ there exists a point $x\in X$ with $d Y (y, f (x)) < ε. Felix Hausdorff ( November 8, 1868 &ndash January 26, 1942) was a German Mathematician who is considered to be one of the founders$

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. In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output

Quasi-isometry is yet another useful generalization. This is a glossary of some terms used in Riemannian geometry and Metric geometry &mdash it doesn't cover the terminology of Differential topology.

Dictionary

-noun

(mathematics) A function between metric spaces (or on a single metric space) having the property that the distance between two images is equal to the distance between their pre-images.

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