NationMaster - Encyclopedia: Axis angle

The axis angle representation of a rotation parameterizes a rotation by two values: an axis, or a line, and an angle describing the magnitude of the rotation about the axis. It is also known as the exponential coordinates of a rotation, for reasons described below. This representation evolves from the fact that any pure rotation in three dimensional space can be represented by an axis, or line in space, and an angle representing the magnitude of the rotation about that axis. A sphere rotating around its axis. ... The axis of rotation of a rotating body is a line such that the distance between any point on the line and any point of the body remains constant under the rotation. ... An angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle. ... A sphere rotating around its axis. ...

1 Uses
- 1.1 Example
2 Relationship to other represenations
3 See also

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.

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 travel Pi/2 radians (or 90 degrees) about the z axis. In axis angle representation, this would be When a circles diameter is 1, its circumference is Ï€. The mathematical constant Ï€ is an irrational real number, approximately equal to 3. ...

$langle mathrm{axis}, mathrm{angle} rangle = left( begin{bmatrix} a_x a_y a_z end{bmatrix},theta right) = left( begin{bmatrix} 0 0 1 end{bmatrix},frac{pi}{2}right)$

Relationship to other represenations

There are many ways to represent a rotation. It is useful to understand how different representation relate to one another, and how to convert between them.

Exponential map from so(3) to SO(3)

The exponential map is used as a transformation from axis angle representation of rotations to rotation matrices. There are two different (but closely related) notions of an exponential map in differential geometry, both of which generalize the ordinary exponential function of mathematical analysis. ... A rotation matrix is a matrix which when multiplied by a vector has the effect of changing the direction of the vector but not its magnitude. ...

$expcolon so(3) to SO(3)$

Essentially, by using a Taylor expansion you can derive a closed form relationship between these two representations. As the degree of the taylor series rises, it approaches the correct function. ...

$R = exp(hat{omega} theta) = sum_{k=0}^inftyfrac{(hat{omega}theta)^k}{k!} = I + hat{omega} theta + frac{1}{2}(hat{omega}theta)^2 + frac{1}{6}(hat{omega}theta)^3 + cdots$

$R = I + hat{omega}left(theta - frac{theta^3}{3!} + frac{theta^5}{5!} - cdotsright) + hat{omega}^2 left(frac{theta^2}{2!} - frac{theta^4}{4!} + frac{theta^6}{6!} - cdotsright)$

$R = I + hat{omega}theta sin(theta) + hat{omega}^2 (1-cos(theta))$

where R is a 3x3 rotation matrix and the hat operator gives the antisymmetric matrix equivalent of the cross product. A rotation matrix is a matrix which when multiplied by a vector has the effect of changing the direction of the vector but not its magnitude. ... In linear algebra, a skew-symmetric (or antisymmetric) matrix is a square matrix A whose transpose is also its negative; that is, it satisfies the equation: AT = −A or in component form, if A = (aij): aij = − aji for all i and j. ... In mathematics, the cross product is a binary operation on vectors in a three-dimensional Euclidean space. ...

Log map from SO(3) to so(3)

To retrieve the axis angle representation of a rotation matrix calculate the angle of rotation A rotation matrix is a matrix which when multiplied by a vector has the effect of changing the direction of the vector but not its magnitude. ...

$theta = arccosleft( frac{mathrm{trace}(R) - 1}{2} right)$

and then use it to find the axis

$omega = frac{1}{2 sin(theta)} begin{bmatrix} r(3,2)-r(2,3) r(1,3)-r(3,1) r(2,1)-r(1,1) end{bmatrix}$

Quaternions

Main article: quaternion

To transform from axis angle coordinates to quaternions use the following expression: In mathematics, the quaternions are a non-commutative extension of the complex numbers. ...

$Q = left(cosleft(frac{theta}{2}right), omega sinleft(frac{theta}{2}right)right)$

Given a unit quaternion, the axis angle coordinates can be extracted using the following:

$theta = 2,arccos(q_0),$

$omega = left{ begin{matrix} frac{q}{ sin( theta/2 ) } , & mathrm{if} ; theta neq 0 0, & mathrm{otherwise} end{matrix} right.$

SO(3) - the group of all rotations in three dimensional space
rotation group - a mathematical look at rotations
homogeneous transformations - a mathematical representation of rigid body motions
screw theory - a representation of rigid body motions and velocities using the concepts of twists, screws and wrenches
Rotation around a fixed axis

Category: Geometry stubs In mechanics and geometry, the rotation group is the set of all rotations of 3-dimensional Euclidean space, R3. ... In mechanics and geometry, the rotation group is the set of all rotations about the origin of 3-dimensional Euclidean space, R3. ... Screw theory was developed by Sir Robert Stawell Ball in 1900, for application in kinematics and statics of mechanisms (rigid body mechanics). ... The simplest three-dimensional case of rotation is rotation of a body about a fixed axis of rotation: each point of the body moves in a plane perpendicular to the axis, carrying out a circular motion, with the circle centered at the intersection of the plane and the axis. ...

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Contents

Example

Relationship to other represenations

Exponential map from so(3) to SO(3)

Log map from SO(3) to so(3)

Quaternions

See also