Isometry

An isometry is in mathematics, a figure that reflects two metric spaces on each other and thereby obtains the metric (distance, distance). That is, the distance between two pixels is the same as that of the prototype points.

In Euclidean geometry and the synthetic especially those isometries are considered that are both geometrical figures for the considered spaces. Usually one speaks of a distance- preserving, length-preserving or isometric mapping. If the required additional properties are clear from the context, simply by an isometry.

Deviating is understood in the Riemannian geometry under an isometry a figure that the Riemannian metric, and thus receives only the lengths of vectors and the lengths of curves. Such an image does not need to maintain the distances between two points.

Definition

If two metric spaces, given, and a picture with the property

Then called isometry from to. Such a map is always injective. Is even bijective, then called isometric isomorphism, and the rooms and hot isometrically isomorphic; otherwise it is called an isometric embedding in.

Special cases

Normed vector spaces

In the normalized vector space is the distance between two vectors is defined by the norm of the difference vector:

Are and two normed vector spaces with norms, respectively, and is a linear map, this map is an isometry if and only if it receives the norm, ie if for all

Applies.

Vector spaces with scalar product

Is a vector space with scalar product, so the induced norm (length ) of a vector is defined as the square root of the scalar product of the vector with itself for the distance between two vectors, and which results in:

And the dot product is referred to here by angle brackets.

Are and vector spaces with scalar product, respectively, and is a linear map, this map is a linear isometry if and only if it receives the scalar product. that is

In the event that the vector spaces and the same, is called such images also orthogonal.

In finite Euclidean vector spaces need not be assumed that the mapping is linear. If the zero vector maps to the zero vector and receives lengths, then it follows from the linearity.

Is an orthonormal basis of such a linear map is an isometry if and only if an orthonormal system is.

The set of all linear isometries of a Euclidean vector space into itself forms a group, the orthogonal group of.

Euclidean point space

A map between two Euclidean point space and is an isometry if and only if there is a linear isometry between the associated Euclidean vector spaces and so

Isometries of a Euclidean point space into itself are called movements.

Other properties

• From the definition it follows immediately that every isometry is continuous.
• Every isometry is even Lipschitz continuous, ie, in particular uniformly continuous. Isometrics are so constantly be continued on the financial statements when the image space is complete.
• Every metric space is isometrically isomorphic to a closed subset of a normed vector space, and each complete metric space is isometrically isomorphic to a closed subset of a Banach space.
• Gilt and and are represented by two figures on each other so the figures are called congruent to each other. Gilt and so they have the same name; otherwise you just talking about isometric drawings.
• Every isometry of Euclidean space also receives angle, surface area and volume.
• General receives any isometry between metric spaces, the Hausdorff dimensions.