In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space...

When used to represent rotation, unit quaternions are also called rotation quaternions as they represent the 3D rotation group. When used to represent...

ceiling Cn is the rotation group of a regular n-sided polygon in 2D and of a regular n-sided pyramid in 3D. If there is e.g. rotational symmetry with respect...

symmetry group is called its rotation group. It is the intersection of its full symmetry group with SO(3), the full rotation group of the 3D space. The...

special orthogonal group of order 4. In this article rotation means rotational displacement. For the sake of uniqueness, rotation angles are assumed to...

v̂, ŵ) form a 3D orthonormal basis. These statements comprise a total of 6 conditions (the cross product contains 3), leaving the rotation matrix with just...

basic 3D rotation (also called elemental rotation) is a rotation about one of the axes of a coordinate system. The following three basic rotation matrices...

(including lattice centering), the point group symmetry operations of reflection, rotation and improper rotation (also called rotoinversion), and the screw...

= − m {\displaystyle n=-m} .) Every proper rotation A {\displaystyle A} in 3D space has an axis of rotation, which is defined such that any vector v {\displaystyle...

under suitable circumstances. For example, the Lie algebra of the 3D rotation group SO(3), [X1, X2] = X3, etc., may be rewritten by a change of variables...

some point (in 3D called the rotation group) all isometries that keep the origin fixed, or more generally, some point (the orthogonal group) all direct isometries...

matrices Circulant matrix Shift operator Quantum Fourier transform 3D rotation group § A note on Lie algebras Brown, Adam R.; Susskind, Leonard (2018-04-25)...

the concept of instant axis of rotation, a line of fixed points. In linear algebra terms, the theorem states that, in 3D space, any two Cartesian coordinate...

elements of a rotation matrix R {\displaystyle R} . The Euler angles form a chart on all of SO(3), the special orthogonal group of rotations in 3D space. The...

Mathematically, a rotation is a map. All rotations about a fixed point form a group under composition called the rotation group (of a particular space)...

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

For example: two 3D figures have mirror symmetry, but with respect to different mirror planes. two 3D figures have 3-fold rotational symmetry, but with...

In mathematics, the special orthogonal group in three dimensions, otherwise known as the rotation group SO(3), is a naturally occurring example of a manifold...

2 {\displaystyle S^{2}} to have symmetry under the action of the 3D rotation group SO(3). That is, by using the a priori knowledge that spheres can be...

Rodrigues' rotation formula. For the notation, see 3D rotation group#A note on Lie algebras. More recently, expressions have appeared for other groups, like...

dynamics. Rotation formalisms in three dimensions Quaternions and spatial rotation Versor Spinors in three dimensions SO(4) 3D rotation group Beltrami–Klein...

geometric correspondence between an axis of rotation and an infinitesimal rotation (see also: 3D rotation group#Lie algebra) around the axis, with speed...

3 ) {\displaystyle {\mathfrak {so}}(3)} , the Lie algebra of the 3d rotation group. Because the Jacobi identity fails in seven dimensions, the seven-dimensional...

}+A_{-}B_{-}^{\star }+A_{0}B_{0}^{\star }} Wigner–Eckart theorem Wigner D matrix 3D rotation group W.J. Thompson (2008). Angular Momentum. John Wiley & Sons. p. 311...

generally contains a subgroup (typically finite) of the 3D rotation group. It may occur that the group {±1} with two elements acts also on the body; this is...

Coxeter group, and like the polyhedral groups of 3D, it can be named by its related convex regular 4-polytope. Related pure rotational groups exist for...

the context of quaternion interpolation for the purpose of animating 3D rotation. It refers to constant-speed motion along a unit-radius great circle...

{\displaystyle M=SO(3),} the special orthogonal group of dimension 3 (otherwise known as 3D rotation group), and P ( M ) = C ∞ ( [ t 0 , t 1 ] , M ) . {\displaystyle...

parameterizes a rotation in a three-dimensional Euclidean space by two quantities: a unit vector e indicating the direction of an axis of rotation, and an angle...

from a point ax,y,z in 3D space to a point bx,y in 2D space looking into the first octant can be written mathematically with rotation matrices as: [ c x c...