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...

distance from a point to a line, in the Euclidean plane The distance from a point to a plane in three-dimensional Euclidean space The distance between two lines...

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...

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

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

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...

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

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...

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

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...

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

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...

sphere geometry Non-Euclidean geometry Noncommutative algebraic geometry Noncommutative geometry Ordered geometry Parabolic geometry Plane geometry Projective...

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

Dual number Dual quaternion Clifford algebra Euclidean plane isometry Affine transformation Projective plane Homogeneous coordinates SLERP Conformal geometric...

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...

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)...

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...

Euclidean space with its usual notion of distance. Other well-known examples are a sphere equipped with the angular distance and the hyperbolic plane...

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...

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...

mathematics, the group of rotations about a fixed point in four-dimensional Euclidean space is denoted SO(4). The name comes from the fact that it is the special...

rotoinversion) is an isometry in Euclidean space that is a combination of a rotation about an axis and a reflection in a plane perpendicular to that...

abstract group Dn, and a common way to visualize it, is the group of Euclidean plane isometries which keep the origin fixed. These groups form one of the two...

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...

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...