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...

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...

two-dimensional or three-dimensional Euclidean space, two geometric figures are congruent if they are related by an isometry; the isometry that relates them is either...

In Euclidean geometry, two-dimensional rotations and reflections are two kinds of Euclidean plane isometries which are related to one another. A rotation...

commonly called respectively Euclidean lines and Euclidean planes. The qualifier "Euclidean" is used to distinguish Euclidean spaces from other spaces that...

the collection of them is then said to form an isometry group of the pseudo-Euclidean space. The isometry group of the subspace of a metric space consisting...

symmetries form a subgroup of the isometry group of the ambient space. This article mainly considers symmetry groups in Euclidean geometry, but the concept may...

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

ISBN 978-3-319-60792-4 Beckman, F. S.; Quarles, D. A. Jr. (1953), "On isometries of Euclidean spaces", Proceedings of the American Mathematical Society, 4 (5):...

a Euclidean space to itself that is an isometry with a hyperplane as the set of fixed points; this set is called the axis (in dimension 2) or plane (in...

a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: For any given line R and point P not on R, in the plane containing...

their symmetry group type. Isometries of the Euclidean plane fall into four categories (see the article Euclidean plane isometry for more information). Translations...

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 of...

UK, the three-bar equal sign ≡ (U+2261) is sometimes used. Euclidean plane isometry Isometry Clapham, C.; Nicholson, J. (2009). "Oxford Concise Dictionary...

two-dimensional Euclidean space, the definite article is used, so the Euclidean plane refers to the whole space. Several notions of a plane may be defined...

of the Euclidean group with in each subset one isometries that keeps the origins fixed, and its combination with all translations. Each isometry is given...

non-Euclidean geometry, the Poincaré half-plane model is a way of representing the hyperbolic plane using points in the familiar Euclidean plane. Specifically...

there exists a quasi-isometry f : M 1 → M 2 {\displaystyle f:M_{1}\to M_{2}} . The map between the Euclidean plane and the plane with the Manhattan distance...

The case of Euclidean and hyperbolic space forms can likewise be reduced to group theory, based on study of the isometry group of Euclidean space and hyperbolic...

coordinates of the points of a Euclidean space of dimension n, En (Euclidean line, E; Euclidean plane, E2; Euclidean three-dimensional space, E3) form...

Fyodorov proved that every periodic tiling of the plane features one of seventeen different groups of isometries. Fyodorov's work marked the unofficial beginning...

two-dimensional direct motion is either a translation or a rotation; see Euclidean plane isometry for details. Rotations in three-dimensional space differ from those...

the isometry group of the sphere (when T is finite), the Euclidean plane (when T has a Z + Z subgroup of finite index), or the hyperbolic plane. Fuchsian...

under glide reflection, it is a motion or isometry. When the context is the two-dimensional Euclidean plane, the hyperplane of reflection is a straight...

mathematics, a non-Euclidean crystallographic group, NEC group or N.E.C. group is a discrete group of isometries of the hyperbolic plane. These symmetry...

of all isometries that leave the origin fixed, or correspondingly, the group of orthogonal matrices. O(3) itself is a subgroup of the Euclidean group E(3)...

three mutually perpendicular planes. More generally, n Cartesian coordinates specify the point in an n-dimensional Euclidean space for any dimension n....

two-dimensional or three-dimensional (i.e., in plane geometry or solid geometry Euclidean spaces). These isometries consist of reflections, rotations, translations...

constant curvature −1. The isometry to the previous models can be realised by stereographic projection from the hyperboloid to the plane { x n + 1 = 0 } {\displaystyle...

Thus, for example, a ball in the Euclidean plane is the same thing as a disk, the area bounded by a circle. In Euclidean 3-space, a ball is taken to be...