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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 under the operation of composition. [ 1 ] By definition, a rotation about the origin is a transformation that preserves the origin, Euclidean distance (so it is an isometry ), and orientation ...
Rotation formalisms are focused on proper (orientation-preserving) motions of the Euclidean space with one fixed point, that a rotation refers to.Although physical motions with a fixed point are an important case (such as ones described in the center-of-mass frame, or motions of a joint), this approach creates a knowledge about all motions.
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 ...
In geometry the rotation group is the group of all rotations about the origin of three-dimensional Euclidean space R 3 under the operation of composition. [1] By definition, a rotation about the origin is a linear transformation that preserves length of vectors (it is an isometry) and preserves orientation (i.e. handedness) of space.
This category deals with topics in physics related to the three-dimensional spherical symmetries of physical objects, including topics concerning rotations in classical mechanics, as well as spin and angular momentum in quantum mechanics, and the representations of the Lie groups SU(2) and SO(3).
Also, unlike the two-dimensional case, a three-dimensional direct motion, in general position, is not a rotation but a screw operation. Rotations about the origin have three degrees of freedom (see rotation formalisms in three dimensions for details), the same as the number of dimensions. A three-dimensional rotation can be specified in a ...
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.
A rotation represented by an Euler axis and angle. In geometry, Euler's rotation theorem states that, in three-dimensional space, any displacement of a rigid body such that a point on the rigid body remains fixed, is equivalent to a single rotation about some axis that runs through the fixed point. It also means that the composition of two ...