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The rotation group is a group under function composition (or equivalently the product of linear transformations). It is a subgroup of the general linear group consisting of all invertible linear transformations of the real 3-space. [2] Furthermore, the rotation group is nonabelian. That is, the order in which rotations are composed makes a ...
An object having symmetry group D n, D nh, or D nd has rotation group D n. An object having a polyhedral symmetry (T, T d, T h, O, O h, I or I h) has as its rotation group the corresponding one without a subscript: T, O or I. The rotation group of an object is equal to its full symmetry group if and only if the object is chiral. In other words ...
Noting that any identity matrix is a rotation matrix, and that matrix multiplication is associative, we may summarize all these properties by saying that the n × n rotation matrices form a group, which for n > 2 is non-abelian, called a special orthogonal group, and denoted by SO(n), SO(n,R), SO n, or SO n (R), the group of n × n rotation ...
The rotation group is a Lie group of rotations about a fixed point. This (common) fixed point or center is called the center of rotation and is usually identified with the origin. The rotation group is a point stabilizer in a broader group of (orientation-preserving) motions. For a particular rotation: The axis of rotation is a line of its ...
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.The various charts on SO(3) set up rival coordinate systems: in this case there cannot be said to be a preferred set of parameters describing a rotation.
A rotation can be represented by a unit-length quaternion q = (w, r →) with scalar (real) part w and vector (imaginary) part r →. The rotation can be applied to a 3D vector v → via the formula = + (+). This requires only 15 multiplications and 15 additions to evaluate (or 18 multiplications and 12 additions if the factor of 2 is done via ...
Let Γ be a finite subgroup of SO(3), the three-dimensional rotation group.There is a natural homomorphism f of SU(2) onto SO(3) which has kernel {±I}. [4] This double cover can be realised using the adjoint action of SU(2) on the Lie algebra of traceless 2-by-2 skew-adjoint matrices or using the action by conjugation of unit quaternions.
With this convention, real vectors correspond to Hermitian matrices. Furthermore, real rotations preserving the form correspond (in the double-valued sense) to unitary matrices of determinant one. In modern terms, this presents the special unitary group SU(2) as a double cover of SO(3). As a consequence, the spinor Hermitian product