Rotation in mathematics is a concept originating in geometry. Any rotation is a motion of a certain space that preserves at least one point. It can describe, for example, the motion of a rigid body around a fixed point. Rotation can have a sign (as in the sign of an angle): a clockwise rotation is a negative magnitude so a counterclockwise turn has a positive magnitude. A rotation is different from other types of motions: translations, which have no fixed points, and (hyperplane) reflections, each of them having an entire (n − 1)-dimensional flat of fixed points in a n-dimensional space. Mathematically, a rotation is a map. All rotations about a fixed point form a group under composition called the rotation group (of a particular space). But in mechanics and, more generally, in physics, this concept is frequently understood as a coordinate transformation (importantly, a transformation of an orthonormal basis), because for any motion of a body there is an inverse transformation which if applied to the frame of reference results in the body being at the same coordinates. For example, in two dimensions rotating a body clockwise about a point keeping the axes fixed is equivalent to rotating the axes counterclockwise about the same point while the body is kept fixed. These two types of rotation are called active and passive transformations.
Related definitions and terminology 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 its fixed points. They exist only in n = 3. The plane of rotation is a plane that is invariant under the rotation. Unlike the axis, its points are not fixed themselves. The axis (where present) and the plane of a rotation are orthogonal. A representation of rotations is a particular formalism, either algebraic or geometric, used to parametrize a rotation map. This meaning is somehow inverse to the meaning in the group theory. Rotations of (affine) spaces of points and of respective vector spaces are not always clearly distinguished. The former are sometimes referred to as affine rotations (although the term is misleading), whereas the latter are vector rotations. See the article below for details.
Definitions and representations
In Euclidean geometry
A motion of a Euclidean space is the same as its isometry: it leaves the distance between any two points unchanged after the transformation. But a (proper) rotation also has to preserve the orientation structure. The "improper rotation" term refers to isometries that reverse (flip) the orientation. In the language of group theory the distinction is expressed as direct vs indirect isometries in the Euclidean group, where the former comprise the identity component. Any direct Euclidean motion can be represented as a composition of a rotation about the fixed point and a translation. In one-dimensional space, there are only trivial rotations. In two dimensions, only a single angle is needed to specify a rotation about the origin – the angle of rotation that specifies an element of the circle group (also known as U(1)). The rotation is acting to rotate an object counterclockwise through an angle θ about the origin; see below for details. Composition of rotations sums their angles modulo 1 turn, which implies that all two-dimensional rotations about the same point commute. Rotations about different points, in general, do not commute. Any two-dimensional direct motion is either a translation or a rotation; see Euclidean plane isometry for details.
Rotations in three-dimensional space differ from those in two dimensions in a number of important ways. Rotations in three dimensions are generally not commutative, so the order in which rotations are applied is important even about the same point. Also, unlike the two-dimensional case, a three-dimensional direct motion, in general position, is not a rotation but a screw operation. Rotations about the origin have three degrees of freedom (see rotation formalisms in three dimensions for details), the same as the number of dimensions. A three-dimensional rotation can be specified in a number of ways. The most usual methods are:
Euler angles (pictured at the left). Any rotation about the origin can be represented as the composition of three rotations defined as the motion obtained by changing one of the Euler angles while leaving the other two constant. They constitute a mixed axes of rotation system because angles are measured with respect to a mix of different reference frames, rather than a single frame that is purely external or purely intrinsic. Specifically, the first angle moves the line of nodes around the external axis z, the second rotates around the line of nodes and the third is an intrinsic rotation (a spin) around an axis fixed in the body that moves. Euler angles are typically denoted as α, β, γ, or φ, θ, ψ. This presentation is convenient only for rotations about a fixed point.
Axis–angle representation (pictured at the right) specifies an angle with the axis about which the rotation takes place. It can be easily visualised. There are two variants to represent it: as a pair consisting of the angle and a unit vector for the axis, or as a Euclidean vector obtained by multiplying the angle with this unit vector, called the rotation vector (although, strictly speaking, it is a pseudovector). Matrices, versors (quaternions), and other algebraic things: see the section Linear and Multilinear Algebra Formalism for details.
A general rotation in four dimensions has only one fixed point, the centre of rotation, and no axis of rotation; see rotations in 4-dimensional Euclidean space for details. Instead the rotation has two mutually orthogonal planes of rotation, each of which is fixed in the sense that points in each plane stay within the planes. The rotation has two angles of rotation, one for each plane of rotation, through which points in the planes rotate. If these are ω1 and ω2 then all points not in the planes rotate through an angle between ω1 and ω2. Rotations in four dimensions about a fixed point have six degrees of freedom. A four-dimensional direct motion in general position is a rotation about certain point (as in all even Euclidean dimensions), but screw operations exist also.
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