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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the axis–angle representation of a rotation parameterizes a
rotation Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
in a
three-dimensional Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called '' parameters'') are required to determine the position of an element (i.e., point). This is the inform ...
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
by two quantities: a
unit vector In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction v ...
indicating the direction of an axis of rotation, and an
angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles ...
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 In mathematics and physics, the right-hand rule is a common mnemonic for understanding orientation of axes in three-dimensional space. It is also a convenient method for quickly finding the direction of a cross-product of 2 vectors. Most of ...
. The rotation axis is sometimes called the Euler axis. It is one of many
rotation formalisms in three dimensions In geometry, various formalisms exist to express a rotation in three dimensions as a mathematical transformation. In physics, this concept is applied to classical mechanics where rotational (or angular) kinematics is the science of quantitative ...
. The axis–angle representation is predicated on
Euler's rotation theorem In geometry, Euler's rotation theorem states that, in three-dimensional space, any displacement of a rigid body such that a point on the rigid body remains fixed, is equivalent to a single rotation about some axis that runs through the fixed p ...
, 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 , \boldsymbol = \theta \mathbf \,. It is used for the
exponential Exponential may refer to any of several mathematical topics related to exponentiation, including: *Exponential function, also: **Matrix exponential, the matrix analogue to the above *Exponential decay, decrease at a rate proportional to value *Expo ...
and
logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 ...
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 (or 90°) about the axis. Viewing the axis-angle representation as an
ordered pair In mathematics, an ordered pair (''a'', ''b'') is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a'') unless ''a'' = ''b''. (In con ...
, this would be ( \mathrm, \mathrm ) = \left( \begin e_x \\ e_y \\ e_z \end,\theta \right) = \left( \begin 0 \\ 0 \\ 1 \end,\frac\right). The above example can be represented as a rotation vector with a magnitude of pointing in the direction, \begin 0 \\ 0 \\ \frac \end.


Uses

The axis–angle representation is convenient when dealing with
rigid body dynamics In the physical science of dynamics, rigid-body dynamics studies the movement of systems of interconnected bodies under the action of external forces. The assumption that the bodies are ''rigid'' (i.e. they do not deform under the action of ...
. It is useful to both characterize
rotation Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
s, and also for converting between different representations of rigid body
motion In physics, motion is the phenomenon in which an object changes its position with respect to time. Motion is mathematically described in terms of displacement, distance, velocity, acceleration, speed and frame of reference to an observer and m ...
, such as homogeneous transformations and twists. When a
rigid body In physics, a rigid body (also known as a rigid object) is a solid body in which deformation is zero or so small it can be neglected. The distance between any two given points on a rigid body remains constant in time regardless of external fo ...
rotates around a fixed axis, its axis–angle data are a constant rotation axis and the rotation angle continuously dependent on
time Time is the continued sequence of existence and event (philosophy), events that occurs in an apparently irreversible process, irreversible succession from the past, through the present, into the future. It is a component quantity of various me ...
. Plugging the three eigenvalues 1 and and their associated three orthogonal axes in a Cartesian representation into
Mercer's theorem In mathematics, specifically functional analysis, Mercer's theorem is a representation of a symmetric positive-definite function on a square as a sum of a convergent sequence of product functions. This theorem, presented in , is one of the most no ...
is a convenient construction of the Cartesian representation of the Rotation Matrix in three dimensions.


Rotating a vector

Rodrigues' rotation formula, named after
Olinde Rodrigues Benjamin Olinde Rodrigues (6 October 1795 – 17 December 1851), more commonly known as Olinde Rodrigues, was a French banker, mathematician, and social reformer. In mathematics Rodrigues is remembered for Rodrigues' rotation formula for vectors ...
, 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 \mathfrak(3) to without computing the full matrix exponential. If is a vector in and is a
unit vector In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction v ...
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 \mathbf_\mathrm = (\cos\theta) \mathbf + (\sin\theta) (\mathbf \times \mathbf) + (1 - \cos\theta) (\mathbf \cdot \mathbf) \mathbf \,. 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 𝔰𝔬(3) to SO(3)

The exponential map effects a transformation from the axis-angle representation of rotations to rotation matrices, \exp\colon \mathfrak(3) \to \mathrm(3) \,. Essentially, by using a
Taylor expansion In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor seri ...
one derives a closed-form relation between these two representations. Given a unit vector \boldsymbol\omega \in \mathfrak(3) = \R^3 representing the unit rotation axis, and an angle, , an equivalent rotation matrix is given as follows, where is the
cross product matrix In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is ...
of , that is, for all vectors , R = \exp(\theta \mathbf) = \sum_^\infty\frac = I + \theta \mathbf + \frac(\theta \mathbf)^2 + \frac(\theta \mathbf)^3 + \cdots Because is skew-symmetric, and the sum of the squares of its above-diagonal entries is 1, the
characteristic polynomial In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The c ...
of is . Since, by the
Cayley–Hamilton theorem In linear algebra, the Cayley–Hamilton theorem (named after the mathematicians Arthur Cayley and William Rowan Hamilton) states that every square matrix over a commutative ring (such as the real or complex numbers or the integers) satisfies ...
, = 0, this implies that \mathbf^3 = -\mathbf \,. 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 R = I + \left(\theta - \frac + \frac - \cdots\right) \mathbf + \left(\frac - \frac + \frac - \cdots\right) \mathbf^2 \,, that is, R = I + (\sin\theta) \mathbf + (1-\cos\theta) \mathbf^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.This holds for the triplet representation of the rotation group, i.e., spin 1. For higher dimensional representations/spins, see 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 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 In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix :R = \begin \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \ ...
, calculate the angle of rotation from the trace of the rotation matrix \theta = \arccos\left( \frac \right) and then use that to find the normalized axis, \boldsymbol = \frac \begin R_-R_ \\ R_-R_ \\ R_-R_ \end ~, where R_ is the component of the rotation matrix, R, in the i-th row and j-th column. Note that the axis-angle representation is not unique since a rotation of -\theta about -\boldsymbol is the same as a rotation of \theta about \boldsymbol . The above calculation of axis vector \omega does not work if is symmetric. For the general case the \omega may be found using null space of , see Rotation matrix#Determining_the_axis. The
matrix logarithm In mathematics, a logarithm of a matrix is another matrix such that the matrix exponential of the latter matrix equals the original matrix. It is thus a generalization of the scalar logarithm and in some sense an inverse function of the matrix exp ...
of the rotation matrix is \log R = \begin 0 & \text \theta = 0 \\ \dfrac \left(R - R^\mathsf\right) & \text \theta \ne 0 \text \theta \in (-\pi, \pi) \end An exception occurs when has
eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...
s equal to . In this case, the log is not unique. However, even in the case where the
Frobenius norm In mathematics, a matrix norm is a vector norm in a vector space whose elements (vectors) are matrices (of given dimensions). Preliminaries Given a field K of either real or complex numbers, let K^ be the -vector space of matrices with m ro ...
of the log is \, \log(R) \, _\mathrm = \sqrt , \theta , \,. Given rotation matrices and , d_g(A,B) := \left\, \log\left(A^\mathsf B\right)\right\, _\mathrm 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 is because these are the first two terms of the Taylor series for .) This formulation also has numerical problems at , 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. R = I + \mathbf \sin\theta + \mathbf^2 (1-\cos\theta) At , we have R = I + 2 \mathbf^2 = I + 2(\boldsymbol \otimes \boldsymbol - I) = 2 \boldsymbol \otimes \boldsymbol - I and so let B := \boldsymbol \otimes \boldsymbol = \frac(R+I) \,, so the diagonal terms of are the squares of the elements of and the signs (up to sign ambiguity) can be determined from the signs of the off-axis terms of .


Unit quaternions

the following expression transforms axis–angle coordinates to
versor In mathematics, a versor is a quaternion of norm one (a ''unit quaternion''). The word is derived from Latin ''versare'' = "to turn" with the suffix ''-or'' forming a noun from the verb (i.e. ''versor'' = "the turner"). It was introduced by Will ...
s (unit
quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quater ...
s): Q = \left(\cos\tfrac, \boldsymbol \sin\tfrac\right) Given a versor represented with its scalar and vector , the axis–angle coordinates can be extracted using the following: \begin \theta &= 2\arccos s \\ px\boldsymbol &= \begin \dfrac, & \text \theta \neq 0 \\ 0, & \text. \end \end A more numerically stable expression of the rotation angle uses the
atan2 In computing and mathematics, the function atan2 is the 2-argument arctangent. By definition, \theta = \operatorname(y, x) is the angle measure (in radians, with -\pi < \theta \leq \pi) between the positive
function: \theta = 2 \operatorname(, \mathbf, ,s)\,, where is the
Euclidean norm Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
of the 3-vector .


See also

*
Homogeneous coordinates In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometr ...
*
Screw theory Screw theory is the algebraic calculation of pairs of vectors, such as forces and moments or angular and linear velocity, that arise in the kinematics and dynamics of rigid bodies. The mathematical framework was developed by Sir Robert Stawe ...
, a representation of rigid body motions and velocities using the concepts of twists, screws and wrenches *
Pseudovector In physics and mathematics, a pseudovector (or axial vector) is a quantity that is defined as a function of some vectors or other geometric shapes, that resembles a vector, and behaves like a vector in many situations, but is changed into its ...
* Rotations without a matrix


References

{{DEFAULTSORT:Axis Angle Representation Rotation in three dimensions Angle