In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed...

In mathematics, the isometry group of a metric space is the set of all bijective isometries (that is, bijective, distance-preserving maps) from the metric...

In mathematics, a quasi-isometry is a function between two metric spaces that respects large-scale geometry of these spaces and ignores their small-scale...

Isometry group Quasi-isometry Dade isometry Euclidean isometry Euclidean plane isometry Itō isometry Isometric (disambiguation) Isometries in physics This...

In mathematics, a Euclidean group is the group of (Euclidean) isometries of a Euclidean space E n {\displaystyle \mathbb {E} ^{n}} ; that is, the transformations...

In mathematics, the Itô isometry, named after Kiyoshi Itô, is a crucial fact about Itô stochastic integrals. One of its main applications is to enable...

{1}{2}}\left(\|x+y\|^{2}-\|x\|^{2}-\|y\|^{2}\right).} An isometry of Euclidean vector spaces is a linear isomorphism. An isometry f : E → F {\displaystyle f\colon E\to F}...

mathematical functional analysis, a partial isometry is a linear map between Hilbert spaces such that it is an isometry on the orthogonal complement of its kernel...

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...

rotation-reflection, rotoreflection, rotary reflection, or rotoinversion) is an isometry in Euclidean space that is a combination of a rotation about an axis and...

in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere. It is a...

In linear algebra, the restricted isometry property (RIP) characterizes matrices which are nearly orthonormal, at least when operating on sparse vectors...

For an object in a metric space, its symmetries form a subgroup of the isometry group of the ambient space. This article mainly considers symmetry groups...

discrete isometry group is an isometry group such that for every point of the metric space the set of images of the point under the isometries is a discrete...

and octahedron. It is the hyperoctahedral group for n = 3. See also the isometries of the cube. With the 4-fold axes as coordinate axes, a fundamental domain...

A fixed point of an isometry group is a point that is a fixed point for every isometry in the group. For any isometry group in Euclidean space the set...

In mathematical finite group theory, the Dade isometry is an isometry from class function on a subgroup H with support on a subset K of H to class functions...

T : V → V′ (isometry) such that Q ( v ) = Q ′ ( T v )  for all  v ∈ V . {\displaystyle Q(v)=Q'(Tv){\text{ for all }}v\in V.} The isometry classes of n-dimensional...

spelled reflexion) is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as the set of fixed points; this set is called the axis...

spacetime, i.e. they are isometries of Minkowski space. They are studied primarily in special relativity. Those isometries that leave the origin fixed...

at p. The lemma allows the exponential map to be understood as a radial isometry, and is of fundamental importance in the study of geodesic convexity and...

If h is a translation, then its conjugation by an isometry can be described as applying the isometry to the translation: the conjugation of a translation...

group (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group of that root system. Specifically, it is the subgroup which is generated...

follows: The Gaussian curvature of a surface is invariant under local isometry. A sphere of radius R has constant Gaussian curvature which is equal to...

direct Euclidean isometry in three dimensions involves a translation and a rotation. The screw displacement representation of the isometry decomposes the...

two-dimensional point group or rosette group is a group of geometric symmetries (isometries) that keep at least one point fixed in a plane. Every such group is a...

isometric linear operators on a given Hilbert space. It states that every isometry is a direct sum of copies of the unilateral shift and a unitary operator...

rigid transformation (also called Euclidean transformation or Euclidean isometry) is a geometric transformation of a Euclidean space that preserves the...

whose isometry group is the complex automorphism group SL(2, R) = Sp(2, R), the Siegel upper half-space has only one metric up to scaling whose isometry group...

surface is called a local isometry. A property of a surface is called an intrinsic property if it is preserved by local isometries and it is called an extrinsic...