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A linear isometry also necessarily preserves angles, therefore a linear isometry transformation is a conformal linear transformation. Examples A linear map from C n {\displaystyle \mathbb {C} ^{n}} to itself is an isometry (for the dot product ) if and only if its matrix is unitary .
In geometry, a Euclidean plane isometry is an isometry of the Euclidean plane, or more informally, a way of transforming the plane that preserves geometrical properties such as length. There are four types: translations , rotations , reflections , and glide reflections (see below § Classification ).
A discrete isometry group is an isometry group such that for every point of the space the set of images of the point under the isometries is a discrete set. In pseudo-Euclidean space the metric is replaced with an isotropic quadratic form ; transformations preserving this form are sometimes called "isometries", and the collection of them is ...
for any point group: the group of all isometries which are a combination of an isometry in the point group and a translation; for example, in the case of the group generated by inversion in the origin: the group of all translations and inversion in all points; this is the generalized dihedral group of R 3, Dih(R 3).
In mathematics, a rigid transformation (also called Euclidean transformation or Euclidean isometry) is a geometric transformation of a Euclidean space that preserves the Euclidean distance between every pair of points. [1] [self-published source] [2] [3] The rigid transformations include rotations, translations, reflections, or any sequence of ...
For example if a crystal has a base-centered Bravais lattice centered on the C face, then a glide of half a cell unit in the a direction gives the same result as a glide of half a cell unit in the b direction. The isometry group generated by just a glide reflection is an infinite cyclic group. Combining two equal glide plane operations gives a ...
In various areas of mathematics, isomorphisms have received specialized names, depending on the type of structure under consideration. For example: An isometry is an isomorphism of metric spaces. A homeomorphism is an isomorphism of topological spaces.
One perhaps non-obvious example of an isometry between spaces described in this article is the map : (,) (,) defined by (,) = (+,). If there is an isometry between the spaces M 1 and M 2 , they are said to be isometric .