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

SO(3) Rotations and reflections in two dimensions CORDIC Infinitesimal rotation matrix Irrational rotation Orientation (geometry) Rodrigues' rotation formula...

rotation can be decomposed into a combination of principal rotations. The combination of any sequence of rotations of an object in three dimensions about...

and reflection is said to have improper rotation symmetry. In 3 dimensions, improper rotation is equivalently defined as a combination of rotation about...

reflections (reflection through the axes of any orthogonal basis); note that orthogonal reflections commute. In 2 dimensions, it is in fact rotation by...

including O(2) itself. Its elements are rotations and reflections, and every such group containing only rotations is a subgroup of the special orthogonal...

two rotations results in another rotation; every rotation has a unique inverse rotation; and the identity map satisfies the definition of a rotation....

describing more complex rotations in four-dimensional space and higher dimensions, where they can be used to break down the rotations into simpler parts....

element). There are only two one-dimensional point groups, the identity group and the reflection group. Point groups in two dimensions, sometimes called rosette...

composition of rotations (i.e. chained rotations). The geometrical definition demonstrates that three consecutive elemental rotations (rotations about the...

(along composite rotations' associative property), the set of all rotations is a group under composition. Every non-trivial rotation is determined by...

groups are: ∞∞, K, or SO(3), all possible rotations. ∞∞m, Kh, or O(3), all possible rotations and reflections. As noted above for the infinite isometry...

a single rotation about some axis that runs through the fixed point. It also means that the composition of two rotations is also a rotation. Therefore...

that it includes orientation-reversing isometries (reflections, glide reflections and improper rotations), as long as those isometries map this particular...

a rotation matrix, as well as the ease of combining successive rotations, make the rotation matrix a useful and popular way to represent rotations, even...

in three dimensions involves a translation and a rotation. The screw displacement representation of the isometry decomposes the translation into two components...

by the rotation, remains in the plane after the rotation. Four-dimensional rotations are of two types: simple rotations and double rotations. A simple...

described as a single reflection, a rotation (2 reflections), or an improper rotation (3 reflections). In four dimensions, double rotations are added that represent...

For rotations in three dimensions, this is the axis of the rotation (a concept that has no meaning in any other dimension). Second, the other two roots...

all rotations about all points implies translational symmetry with respect to all translations (because translations are compositions of rotations about...

isometries): translations, rotations, reflections and glide reflections. Translating a set of points of the plane, preserving the distances and directions between...

Rotational spherical symmetry is isomorphic with the rotation group SO(3), and can be parametrized by the Davenport chained rotations pitch, yaw, and...

such as length. There are four types: translations, rotations, reflections, and glide reflections (see below § Classification). The set of Euclidean plane...

Coordinate rotations and reflections Householder transformation Inversive geometry Plane of rotation Reflection mapping Reflection group Reflection symmetry...

In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the...

symmetry is symmetry with respect to some or all rotations in m-dimensional Euclidean space. Rotations are direct isometries, i.e., isometries preserving...

Dn contains rotations only, not reflections. The other group is pyramidal symmetry Cnv of the same order, 2n. With reflection symmetry in a plane perpendicular...

}Q{\mathbf {v} }.} Thus finite-dimensional linear isometries—rotations, reflections, and their combinations—produce orthogonal matrices. The converse...

reflections in two perpendicular directions, and a rotation of order two (180°) whose centre is not on a reflection axis. It also has two rotations whose...

group E(n) comprises all translations, rotations, and reflections of E n {\displaystyle \mathbb {E} ^{n}} ; and arbitrary finite combinations of them....