Rotations

Rotations are orientation-preserving isometries of a space, meaning they preserve distances between points. In two dimensions, a rotation moves every point around a fixed center by a given angle. In three dimensions, a rotation is a rigid motion that preserves distances and orientation, which can be described as turning about an axis through a point or, more generally, by a rotation matrix with determinant 1.

Common representations include: in 2D, a rotation by angle theta about the origin is given by the matrix [[cos theta, -sin theta], [sin theta, cos theta]]; in 3D, rotations are represented by 3x3 orthogonal matrices with determinant 1, or equivalently by an axis-angle specification, Euler angles, or a unit quaternion.

Rotations compose by matrix multiplication; the result of applying two rotations is another rotation. The set of all rotations in the plane about a fixed point is abelian; in three dimensions, composition is generally non-commutative. The collection of all n-dimensional rotations forms the special orthogonal group SO(n).

Rotations are central in physics and engineering: angular momentum and rotational symmetry, attitude representation in robotics, and object orientation in computer graphics. They enable coordinate transformations between reference frames and are used in astronomy, navigation, and imaging. Quaternions are often preferred in computations to avoid issues such as gimbal lock associated with Euler angles.

Rotations can also be defined about an arbitrary point by combining a translation, a rotation about the origin, and the inverse translation. In higher dimensions, rotations form the group SO(n) and are distinct from improper rotations, which include reflections (determinant -1).