Rotation group

The rotation group of an object is the symmetry group within E+(n), the group of direct isometries; in other words, the intersection of the full symmetry group and the group of direct isometries. For chiral objects it is the same as the full symmetry group.

Related Topics:
Symmetry group - Group of direct isometries - Chiral

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:The rest of the article is about a different but related meaning.

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In mechanics and geometry, the rotation group is the set of all rotations of 3-dimensional Euclidean space R3 which keep the origin fixed. It is a subgroup of E+(n), the group of direct isometries (which do not keep a point fixed).

Related Topics:
Mechanics - Geometry - Rotation - Euclidean space - Group of direct isometries

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Such a rotation is a linear transformation that preserves the length of vectors, and also preserves the orientation, or handedness, of space. A transformation that preserves length but reverses orientation is sometimes called an improper rotation. In such a context, the term proper rotation is used if orientation is preserved.

Related Topics:
Linear transformation - Vector - Orientation - Improper rotation

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The composition of two rotations is a rotation, and every rotation has a unique inverse which is again a rotation. These properties give the set of all rotations the mathematical structure of a group with composition as the group operation. It so happens that this group has a natural manifold structure for which the group operations are smooth, so that the rotation group is actually a real Lie group. This group is often denoted SO(3) for reasons that are explained below.

Related Topics:
Composition - Group - Manifold - Smooth - Lie group

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The group should not be confused with the larger group of all moves of a rigid body: each of these is a combination of a rotation about an arbitrary axis and a translation along the axis, or put differently, a combination of an element of SO(3) and an arbitrary translation.

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