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).
The set of Euclidean plane isometries forms a group under composition: the Euclidean group in two dimensions. It is generated by reflections in lines, and every element of the Euclidean group is the composite of at most three distinct reflections.
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In geometry, a Euclideanplaneisometry is an isometry of the Euclideanplane, 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 Euclideanplaneisometries which are related to one another. A rotation...
commonly called respectively Euclidean lines and Euclideanplanes. 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...
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...
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):...
mathematical object covering a whole Euclideanplane by repeating a motif indefinitely, in manner that certain isometries keep the drawing unchanged. For each...
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 the Euclidean group with in each subset one isometries that keeps the origins fixed, and its combination with all translations. Each isometry is given...
under glide reflection, it is a motion or isometry. When the context is the two-dimensional Euclideanplane, the hyperplane of reflection is a straight...
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 isometry group of the sphere (when T is finite), the Euclideanplane (when T has a Z + Z subgroup of finite index), or the hyperbolic plane. Fuchsian...
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....
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...
coordinates of the points of a Euclidean space of dimension n, En (Euclidean line, E; Euclideanplane, E2; Euclidean three-dimensional space, E3) form...
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