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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 ...
For a bounded object, the proper 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 rotation group of a bounded object is equal to its full symmetry group if and only if the object is chiral .
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.
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
Rotations and translations are both special cases of screw motions, e.g. a rotation around a line in space followed by a translation directed along the same line. This group is usually called SE(3), the group of Special (handedness-preserving) Euclidean (distance-preserving) transformations in 3 dimensions.
For m = 3, this is the rotation group SO(3). [19] Phrased slightly differently, the rotation group of an object is the symmetry group within E + (m), the group of rigid motions; [20] that is, the intersection of the full symmetry group and the group of rigid motions. For chiral objects, it is the same as the full symmetry group.
It also means that the composition of two rotations is also a rotation. Therefore the set of rotations has a group structure, known as a rotation group. The theorem is named after Leonhard Euler, who proved it in 1775 by means of spherical geometry. The axis of rotation is known as an Euler axis, typically represented by a unit vector ê.
3D visualization of a sphere and a rotation about an Euler axis (^) by an angle of In 3-dimensional space, according to Euler's rotation theorem, any rotation or sequence of rotations of a rigid body or coordinate system about a fixed point is equivalent to a single rotation by a given angle about a fixed axis (called the Euler axis) that runs through the fixed point. [6]