Axis–angle representation


In mathematics, the axis–angle representation of a rotation parameterizes a rotation in a three-dimensional Euclidean space by two quantities: a unit vector indicating the direction of an axis of rotation, and an angle describing the magnitude of the rotation about the axis. Only two numbers, not three, are needed to define the direction of a unit vector rooted at the origin because the magnitude of is constrained. For example, the elevation and azimuth angles of 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.
It is one of many rotation formalisms in three dimensions. 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.

Rotation vector

The axis–angle representation is equivalent to the more concise rotation vector, also called the Euler vector. 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,
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, for any integer, encodes exactly the same rotation as a rotation vector of length. Thus, there are at least a countable infinity of rotation vectors corresponding to any rotation. Furthermore, all rotations by are the same as no rotation at all, so, for a given integer, all rotation vectors of length, 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 direction. Then if you turn to your left, you will rotate radians about the axis. Viewing the axis-angle representation as an ordered pair, this would be
The above example can be represented as a rotation vector with a magnitude of pointing in the direction,

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 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.

Rotating a vector

, named after Olinde Rodrigues, is an efficient algorithm for rotating a Euclidean vector, given a rotation axis and an angle of rotation. In other words, Rodrigues' formula provides an algorithm to compute the exponential map from to without computing the full matrix exponential.
If is a vector in and is a unit vector rooted at the origin describing an axis of rotation about which is rotated by an angle, Rodrigues' rotation formula to obtain the rotated vector is
For the rotation of a single vector it may be more efficient than converting and into a rotation matrix to rotate the vector.

Relationship to other representations

There are several ways to represent a rotation. It is useful to understand how different representations relate to one another, and how to convert between them. Here the unit vector is denoted instead of.

Exponential map from \mathfrak{so}(3) to SO(3)

The exponential map effects a transformation from the axis-angle representation of rotations to rotation matrices,
Essentially, by using a Taylor expansion one derives a closed-form relation between these two representations. Given a unit vector representing the unit rotation axis, and an angle,, an equivalent rotation matrix is given as follows, where is the cross product matrix of, that is, for all vectors,
Because is skew-symmetric, and the sum of the squares of its above-diagonal entries is 1, the characteristic polynomial of is. Since, by the Cayley–Hamilton theorem, = 0, this implies that
As a result,,,,.
This cyclic pattern continues indefinitely, and so all higher powers of can be expressed in terms of and. Thus, from the above equation, it follows that
that is,
by the Taylor series formula for trigonometric functions.
This is a Lie-algebraic derivation, in contrast to the geometric one in the article Rodrigues' rotation formula.
Due to the existence of the above-mentioned exponential map, the unit vector representing the rotation axis, and the angle are sometimes called the exponential coordinates of the rotation matrix.

Log map from SO(3) to \mathfrak{so}(3)

Let continue to denote the 3 × 3 matrix that effects the cross product with the rotation axis : for all vectors in what follows.
To retrieve the axis–angle representation of a rotation matrix, calculate the angle of rotation from the trace of the rotation matrix
and then use that to find the normalized axis,
The Matrix logarithm of the rotation matrix is
An exception occurs when has eigenvalues equal to. In this case, the log is not unique. However, even in the case where θ = the Frobenius norm of the log is
Given rotation matrices and,
is the geodesic distance on the 3D manifold of rotation matrices.
For small rotations, the above computation of may be numerically imprecise as the derivative of arccos goes to infinity as. In that case, the off-axis terms will actually provide better information about since, for small angles,.
This formulation also has numerical problems at, where the off-axis terms do not give information about the rotation axis. In that case, we must reconsider the above formula.
At, we have
and so let
so the diagonal terms of are the squares of the elements of and the signs can be determined from the signs of the off-axis terms of .

Unit quaternions

the following expression transforms axis–angle coordinates to versors :
Given a versor represented with its scalar and vector, the axis–angle coordinates can be extracted using the following:
A more numerically stable expression of the rotation angle uses the atan2 function:
where is the Euclidean norm of the 3-vector.