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Axis–angle representation

Inmathematics, the axis–angle representation parameterizes a rotation in a three-dimensional Euclidean space by two quantities: a unit vector e indicating the direction (geometry) 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.

ByRodrigues' 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[edit]

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 θ, $${\displaystyle {\boldsymbol {\theta }}=\theta \mathbf {e} \,.}$$ 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[edit]

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 $${\displaystyle (\mathrm {axis} ,\mathrm {angle} )=\left({\begin{bmatrix}e_{x}\\e_{y}\\e_{z}\end{bmatrix}},\theta \right)=\left({\begin{bmatrix}0\\0\\-1\end{bmatrix}},{\frac {-\pi }{2}}\right).}$$

The above example can be represented as a rotation vector with a magnitude of ⁠π/2⁠ pointing in the z direction, $${\displaystyle {\begin{bmatrix}0\\0\\{\frac {\pi }{2}}\end{bmatrix}}.}$$

Uses[edit]

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^{[clarification needed]} 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 dependentontime.

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.

Rotating a vector[edit]

Rodrigues' rotation formula, 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 ${\displaystyle {\mathfrak {so}}($3)} toSO(3) without computing the full matrix exponential.

Ifv is a vector in R³ and e is a unit vector rooted at the origin describing an axis of rotation about which v is rotated by an angle θ, Rodrigues' rotation formula to obtain the rotated vector is $${\displaystyle \mathbf {v} _{\mathrm {rot} }=(\cos \theta )\mathbf {v} +(\sin \theta )(\mathbf {e} \times \mathbf {v} )+(1-\cos \theta )(\mathbf {e} \cdot \mathbf {v} )\mathbf {e} \,.}$$

For the rotation of a single vector it may be more efficient than converting e and θ into a rotation matrix to rotate the vector.

Relationship to other representations[edit]

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 ω instead of e.

Exponential map from 𝔰𝔬(3) to SO(3)[edit]

The exponential map effects a transformation from the axis-angle representation of rotations to rotation matrices, $${\displaystyle \exp \colon {\mathfrak {so}}($$3)\to \mathrm {SO} (3)\,.}

Essentially, by using a Taylor expansion one derives a closed-form relation between these two representations. Given a unit vector ${\textstyle {\boldsymbol {\omega }}\in {\mathfrak {so}}($3)=\mathbb {R} ^{3}} representing the unit rotation axis, and an angle, θ ∈ R, an equivalent rotation matrix R is given as follows, where K is the cross product matrixofω, that is, Kv = ω × v for all vectors v ∈ R³, $${\displaystyle R=\exp(\theta \mathbf {K} )=\sum _{k=0}^{\infty }{\frac {(\theta \mathbf {K} )^{k}}{k!}}=I+\theta \mathbf {K} +{\frac {1}{2!}}(\theta \mathbf {K} )^{2}+{\frac {1}{3!}}(\theta \mathbf {K} )^{3}+\cdots }$$

Because K is skew-symmetric, and the sum of the squares of its above-diagonal entries is 1, the characteristic polynomial P(t)ofKisP(t) = det(K − tI) = −(t³ + t). Since, by the Cayley–Hamilton theorem, P(K) = 0, this implies that $${\displaystyle \mathbf {K} ^{3}=-\mathbf {K} \,.}$$ As a result, K⁴ = –K², K⁵ = K, K⁶ = K², K⁷ = –K.

This cyclic pattern continues indefinitely, and so all higher powers of K can be expressed in terms of K and K². Thus, from the above equation, it follows that $${\displaystyle R=I+\left(\theta -{\frac {\theta ^{3}}{3!}}+{\frac {\theta ^{5}}{5!}}-\cdots \right)\mathbf {K} +\left({\frac {\theta ^{2}}{2!}}-{\frac {\theta ^{4}}{4!}}+{\frac {\theta ^{6}}{6!}}-\cdots \right)\mathbf {K} ^{2}\,,}$$ that is, $${\displaystyle R=I+(\sin \theta )\mathbf {K} +(1-\cos \theta )\mathbf {K} ^{2}\,,}$$

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.^[1]

Due to the existence of the above-mentioned exponential map, the unit vector ω representing the rotation axis, and the angle θ are sometimes called the exponential coordinates of the rotation matrix R.

Log map from SO(3) to 𝔰𝔬(3)[edit]

Let K continue to denote the 3 × 3 matrix that effects the cross product with the rotation axis ω: K(v) = ω × v for all vectors v 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: $${\displaystyle \theta =\arccos \left({\frac {\operatorname {Tr} ($$R)-1}{2}}\right)} and then use that to find the normalized axis, $${\displaystyle {\boldsymbol {\omega }}={\frac {1}{2\sin \theta }}{\begin{bmatrix}R_{32}-R_{23}\\R_{13}-R_{31}\\R_{21}-R_{12}\end{bmatrix}}~,}$$

where ${\displaystyle R_{ij}}$ is the component of the rotation matrix, ${\displaystyle R}$, in the ${\displaystyle i}$-th row and ${\displaystyle j}$-th column.

The axis-angle representation is not unique since a rotation of ${\displaystyle -\theta }$ about ${\displaystyle -{\boldsymbol {\omega }}}$ is the same as a rotation of ${\displaystyle \theta }$ about ${\displaystyle {\boldsymbol {\omega }}}$.

The above calculation of axis vector ${\displaystyle \omega }$ does not workifR is symmetric. For the general case the ${\displaystyle \omega }$ may be found using null space of R-I, see rotation matrix#Determining the axis.

The matrix logarithm of the rotation matrix Ris$${\displaystyle \log R={\begin{cases}0&{\text{if }}\theta =0\\{\dfrac {\theta }{2\sin \theta }}\left(R-R^{\mathsf {T}}\right)&{\text{if }}\theta \neq 0{\text{ and }}\theta \in (-\pi ,\pi )\end{cases}}}$$

An exception occurs when R has eigenvalues equal to −1. In this case, the log is not unique. However, even in the case where θ = π the Frobenius norm of the log is $${\displaystyle \|\log($$R)\|_{\mathrm {F} }={\sqrt {2}}|\theta |\,.} Given rotation matrices A and B, $${\displaystyle d_{g}(A,B):=\left\|\log \left(A^{\mathsf {T}}B\right)\right\|_{\mathrm {F} }}$$ is the geodesic distance on the 3D manifold of rotation matrices.

For small rotations, the above computation of θ may be numerically imprecise as the derivative of arccos goes to infinity as θ → 0. In that case, the off-axis terms will actually provide better information about θ since, for small angles, R ≈ I + θK. (This is because these are the first two terms of the Taylor series for exp(θK).)

This formulation also has numerical problems at θ = π, where the off-axis terms do not give information about the rotation axis (which is still defined up to a sign ambiguity). In that case, we must reconsider the above formula.

$${\displaystyle R=I+\mathbf {K} \sin \theta +\mathbf {K} ^{2}(1-\cos \theta )}$$Atθ = π, we have $${\displaystyle R=I+2\mathbf {K} ^{2}=I+2({\boldsymbol {\omega }}\otimes {\boldsymbol {\omega }}-I)=2{\boldsymbol {\omega }}\otimes {\boldsymbol {\omega }}-I}$$ and so let $${\displaystyle B:={\boldsymbol {\omega }}\otimes {\boldsymbol {\omega }}={\frac {1}{2}}(R+I)\,,}$$ so the diagonal terms of B are the squares of the elements of ω and the signs (up to sign ambiguity) can be determined from the signs of the off-axis terms of B.

Unit quaternions[edit]

The following expression transforms axis–angle coordinates to versors (unit quaternions): $${\displaystyle \mathbf {q} =\left(\cos {\tfrac {\theta }{2}},{\boldsymbol {\omega }}\sin {\tfrac {\theta }{2}}\right)}$$

Given a versor q = r + v represented with its scalar r and vector v, the axis–angle coordinates can be extracted using the following: $${\displaystyle {\begin{aligned}\theta &=2\arccos r\\[8px]{\boldsymbol {\omega }}&={\begin{cases}{\dfrac {\mathbf {v} }{\sin {\tfrac {\theta }{2}}}},&{\text{if }}\theta \neq 0\\0,&{\text{otherwise}}.\end{cases}}\end{aligned}}}$$

A more numerically stable expression of the rotation angle uses the atan2 function: $${\displaystyle \theta =2\operatorname {atan2} (|\mathbf {v} |,r)\,,}$$ where |v| is the Euclidean norm of the 3-vector v.

See also[edit]

• Homogeneous coordinates
• Pseudovector
• Rotations without a matrix
• Screw theory, a representation of rigid-body motions and velocities using the concepts of twists, screws, and wrenches

References[edit]