In mathematics, the axis–angle representation parameterizes a rotation in a three-dimensional Euclidean space by two quantities: a unit vector e indicating the direction of an axis of rotation, and an angle of rotation θ describing the magnitude and sense (e.g., clockwise) of the rotation about the axis. Only two numbers, not three, are needed to define the direction of a unit vector e rooted at the origin because the magnitude of e is constrained. For example, the elevation and azimuth angles of e suffice to locate it in any particular Cartesian coordinate frame. By Rodrigues' rotation formula, the angle and axis determine a transformation that rotates three-dimensional vectors. The rotation occurs in the sense prescribed by the right-hand rule. The rotation axis is sometimes called the Euler axis. The axis–angle representation is predicated on Euler's rotation theorem, which dictates that any rotation or sequence of rotations of a rigid body in a three-dimensional space is equivalent to a pure rotation about a single fixed axis. It is one of many rotation formalisms in three dimensions.
Rotation vector The axis–angle representation is equivalent to the more concise rotation vector, also called the Euler vector (not to be confused with a vector of Euler angles). In this case, both the rotation axis and the angle are represented by a vector codirectional with the rotation axis whose length is the rotation angle θ,
θ = θ e . {\displaystyle {\boldsymbol {\theta }}=\theta \mathbf {e} \,.}
It is used for the exponential and logarithm maps involving this representation. Many rotation vectors correspond to the same rotation. In particular, a rotation vector of length θ + 2πM, for any integer M, encodes exactly the same rotation as a rotation vector of length θ. Thus, there are at least a countable infinity of rotation vectors corresponding to any rotation. Furthermore, all rotations by 2πM are the same as no rotation at all, so, for a given integer M, all rotation vectors of length 2πM, in all directions, constitute a two-parameter uncountable infinity of rotation vectors encoding the same rotation as the zero vector. These facts must be taken into account when inverting the exponential map, that is, when finding a rotation vector that corresponds to a given rotation matrix. The exponential map is onto but not one-to-one.
Example Say you are standing on the ground and you pick the direction of gravity to be the negative z direction. Then if you turn to your left, you will rotate -π/2 radians (or -90°) about the -z axis. Viewing the axis-angle representation as an ordered pair, this would be
( a x i s , a n g l e ) = ( [ e x e y e z ] , θ ) = ( [ 0 0 − 1 ] , − π 2 ) . {\displaystyle (\mathrm {axis} ,\mathrm {angle} )=\left({\begin{bmatrix}e_{x}\\e_{y}\\e_{z}\end{bmatrix}},\theta \right)=\left({\begin{bmatrix}0\\0\\-1\end{bmatrix}},{\frac {-\pi }{2}}\right).}
The above example can be represented as a rotation vector with a magnitude of π/2 pointing in the z direction,
[ 0 0 π 2 ] . {\displaystyle {\begin{bmatrix}0\\0\\{\frac {\pi }{2}}\end{bmatrix}}.}
Uses The axis–angle representation is convenient when dealing with rigid-body dynamics. It is useful to both characterize rotations, and also for converting between different representations of rigid body motion, such as homogeneous transformations and twists. When a rigid body rotates around a fixed axis, its axis–angle data are a constant rotation axis and the rotation angle continuously dependent on time. Plugging the three eigenvalues 1 and e±iθ and their associated three orthogonal axes in a Cartesian representation into Mercer's theorem is a convenient construction of the Cartesian representation of the Rotation Matrix in three dimensions.
Application
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