Rotation group Totally Explained

Everything about Spherical Symmetry totally explained

In mechanics and geometry, the rotation group is the group of all rotations about the origin of 3-dimensional Euclidean space R³ under the operation of composition. By definition, a rotation about the origin is a linear transformation that preserves length of vectors and preserves orientation (for example handedness) of space. A length-preserving transformation which reverses orientation is called an improper rotation.
Composing two rotations results in another rotation; every rotation has a unique inverse rotation; and the identity map satisfies the definition of a rotation. Owing to the above properties, the set of all rotations is a group under composition. Moreover, the rotation group has a natural manifold structure for which the group operations are smooth; so it's in fact a Lie group. The rotation group is often denoted SO(3) for reasons explained below.

Properties

Besides just preserving length, rotations also preserve the angles between vectors. This follows from the fact that the standard dot product between two vectors u and v can be written purely in terms of length: ť

mathbf. Topologically, this map is a two-to-one covering map .

Lie algebra

Since SO(3) is a Lie subgroup of the general linear group GL(3), its Lie algebra can identified with a Lie subalgebra of gl(3), the algebra of 3×3 matrices with the commutator given by ť

[A,B] = AB - BA.

The condition that a matrix A belong to SO(3) is that ť (*)

AA^T = I.

tmapsto A(t)

is a one-parameter subgroup of SO(3), then differentiating (*) with respect to t gives ť

A'(0) + A'(0)^T = 0

and so the Lie algebra so(3) consists of all skew-symmetric 3×3 matrices.

Representations of rotations

We have seen that there are a variety of ways to represent rotations:

as orthogonal matrices with determinant 1,

by axis and rotation angle

via the unit quaternions (see quaternions and spatial rotations) and the map S³ → SO(3). Another method is to specify an arbitrary rotation by a sequence of rotations about some fixed axes. See:

Euler angles See charts on SO(3) for further discussion.

Generalizations

The rotation group generalizes quite naturally to n-dimensional Euclidean space, Rⁿ. The group of all proper and improper rotations in n dimensions is called the orthogonal group, O(n), and the subgroup of proper rotations is called the special orthogonal group, SO(n).
   In special relativity, one works in a 4-dimensional vector space, known as Minkowski space rather than 3-dimensional Euclidean space. Unlike Euclidean space, Minkowski space has an inner product with an indefinite signature. However, one can still define generalized rotations which preserve this inner product. Such generalized rotations are known as Lorentz transformations and the group of all such transformations is called the Lorentz group.
   The rotation group SO(3) can be described as a subgroup of E⁺(3), the Euclidean group of direct isometries of R³. This larger group is the group of all motions 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.
   In general, the rotation group of an object is the symmetry group within 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's the same as the full symmetry group.

Further Information

Get more info on 'Spherical Symmetry'.

External Link Exchanges

Do you know how hard it is to get a link from a large encyclopaedia? Well we're different and will prove it. To get a link from us just add the following HTML to your site on a relevant page:

<a href="http://rotation_group.totallyexplained.com">Rotation group Totally Explained</a>

Then simply click through this link from your web page. Our crawlers will verify your link, extract the title of your web page and instantly add a link back to it. If you like you can remove the words Totally Explained and embed the link in article text.
As long as your link remains in place, we'll keep our link to you right here. Please play fair - our crawlers are watching. Your site must be closely related to this one's topic. Any kind of spamming, dubious practises or removing the link will result in your link from us being dropped and, potentially, your whole site being banned.