3D rotation group

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 R 3 {\displaystyle \mathbb {R} ^{3}} under the operation of composition, which combines two rotations by performing one after the other. A rotation about a point is a transformation that preserves that point, while also preserving the Euclidean distance between any two points (so it is an isometry), and orientation (i.e., handedness of space). 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 (along composite rotations' associative property), the set of all rotations is a group under composition. Every non-trivial rotation is determined by its axis of rotation (a line through the origin) and its angle of rotation. Rotations are not commutative (for example, rotating R 90° in the x-y plane followed by S 90° in the y-z plane is not the same as S followed by R), making the 3D rotation group a nonabelian group. Moreover, the rotation group has a natural structure as a manifold for which the group operations are smoothly differentiable, so it is in fact a Lie group. It is compact and has dimension 3. Rotations are linear transformations of R 3 {\displaystyle \mathbb {R} ^{3}} and can therefore be represented by matrices once a basis (the three orthogonal unit vectors of the x, y, and z axes) of R 3 {\displaystyle \mathbb {R} ^{3}} has been chosen. Specifically, if we choose an orthonormal basis of R 3 {\displaystyle \mathbb {R} ^{3}} , every rotation is described by an orthogonal 3 × 3 matrix (i.e., a 3 × 3 matrix with real entries which, when multiplied by its transpose, results in...