Rotation, also known as rotational motion or rotary motion, is the movement of an object that leaves at least one point unchanged. In 2 dimensions, a plane figure can rotate in either a clockwise or counterclockwise sense around a point called the center of rotation. In 3 dimensions, a solid figure rotates around an imaginary line called an axis of rotation. The special case of a rotation with an internal axis passing through the body's own center of mass is known as a spin (or autorotation). In that case, the surface intersection of the internal spin axis can be called a pole; for example, Earth's rotation defines the geographical poles. A rotation around an axis completely external to the moving body is called a revolution (or orbit), as in a planetary orbit, e.g., Earth's orbit around the Sun. The ends of the external axis of revolution can be called the orbital poles. Either type of rotation is involved in a corresponding type of angular velocity (spin angular velocity and orbital angular velocity) and angular momentum (spin angular momentum and orbital angular momentum).
Mathematics
Mathematically, a rotation is a rigid body movement which, unlike a translation, keeps at least one point fixed. This definition applies to rotations in two dimensions (in a plane), in which exactly one point is kept fixed; and also in three dimensions (in space), in which additional points may be kept fixed (as in rotation around a fixed axis, as infinite line). All rigid body movements are rotations, translations, or combinations of the two. A rotation is simply a progressive radial orientation to a common point. That common point lies within the axis of that motion. The axis is perpendicular to the plane of the motion. If a rotation around a point or axis is followed by a second rotation around the same point/axis, a third rotation results. The reverse (inverse) of a rotation is also a rotation. Thus, the rotations around a point/axis form a group. However, a rotation around a point or axis and a rotation around a different point/axis may result in something other than a rotation, e.g. a translation. Rotations around the x, y and z axes are called principal rotations. Rotation around any axis can be performed by taking a rotation around the x axis, followed by a rotation around the y axis, and followed by a rotation around the z axis. That is to say, any spatial rotation can be decomposed into a combination of principal rotations.
Fixed axis vs. fixed point The combination of any sequence of rotations of an object in three dimensions about a fixed point is always equivalent to a rotation about an axis (which may be considered to be a rotation in the rotation plane that is perpendicular to that axis). Similarly, the rotation rate of an object in three dimensions at any instant is about some axis, although this axis may be changing over time. In other than three dimensions, it does not make sense to describe a rotation as being around an axis, since more than one axis through the object may be kept fixed; instead, simple rotations are described as being in a plane. In four or more dimensions, a combination of two or more rotations about a plane is not in general a rotation in a single plane.
Axis of 2-dimensional rotations
2-dimensional rotations, unlike the 3-dimensional ones, possess no axis of rotation, only a point about which the rotation occurs. This is equivalent, for linear transformations, with saying that there is no direction in the plane which is kept unchanged by a 2-dimensional rotation, except, of course, the identity. The question of the existence of such a direction is the question of existence of an eigenvector for the matrix A representing the rotation. Every 2D rotation around the origin through an angle θ {\displaystyle \theta } in counterclockwise direction can be quite simply represented by the following matrix:
A = [ cos θ − sin θ sin θ cos θ ] {\displaystyle A={\begin{bmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \end{bmatrix}}}
A standard eigenvalue determination leads to the characteristic equation
λ 2 − 2 λ cos θ + 1 = 0 , {\displaystyle \lambda ^{2}-2\lambda \cos \theta +1=0,}
which has
cos θ ± i sin θ {\displaystyle \cos \theta \pm i\sin \theta }
as its eigenvalues. Therefore, there is no real eigenvalue whenever cos θ ≠ ± 1 {\displaystyle \cos \theta \neq \pm 1} , meaning that no real vector in the plane is kept unchanged by A.
Rotation angle and axis in 3 dimensions
Knowing that the trace is an invariant, the rotation angle α {\displaystyle \alpha } for a proper orthogonal 3×3 rotation matrix A {\displaystyle A} is found by
α = cos − 1 ( A 11 + A 22 + A 33 − 1 2 ) {\displaystyle \alpha =\cos ^{-1}\left({\frac {A_{11}+A_{22}+A_{33}-1}{2}}\right)}
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