- Euler's rotation theorem
In

kinematics ,**Euler's rotation theorem**states that, inthree-dimensional space , any displacement of a rigid body such that a point on the rigid body remains fixed, is equivalent to a rotation about a fixed axis through that point. The theorem is named afterLeonhard Euler .In

mathematical terms, this is a statement that, in 3D space, any two coordinate systems with a common origin are related by a rotation about some fixed axis. This also means that the product of two rotation matrices is again a rotation matrix. A (non-identity)rotation matrix has a realeigenvalue which is equal to unity. Theeigenvector corresponding to this eigenvalue is the axis of rotation connecting the two systems.**Applications****Generators of rotations**Suppose we specify an axis of rotation by a unit vector ["x", "y", "z"] , and suppose we have an infinitely small rotation of angle Δθ about that axis. To first order the rotation matrix ΔR is represented as:

:$Delta\; R\; =\; egin\{bmatrix\}\; 100\backslash \; 010\backslash \; 001\; end\{bmatrix\}+\; egin\{bmatrix\}\; 0\; z-y\backslash \; -z\; 0\; x\backslash \; y\; -x\; 0\; end\{bmatrix\},Delta\; heta=\; mathbf\{I\}+mathbf\{A\},Delta\; heta.$

A finite rotation through angle θ about this axis may be seen as a succession of small rotations about the same axis. Approximating Δθ as θ/"N" where "N" is a large number, a rotation of θ about the axis may be represented as:

:$R\; =left(mathbf\{1\}+frac\{mathbf\{A\}\; heta\}\{N\}\; ight)^Napprox\; e^\{mathbf\{A\}\; heta\}.$

It can be seen that Euler's theorem essentially states that

__all__rotations may be represented in this form. The product $mathbf\{A\}\; heta$ is the "generator" of the particular rotation. Analysis is often easier in terms of these generators, rather than the full rotation matrix. Analysis in terms of the generators is known as theLie algebra of the rotation group.**Quaternions**It follows from Euler's theorem that the relative orientation of any pair of coordinate systems may be specified by a set of four numbers. Three of these numbers are the direction cosines that orient the eigenvector. The fourth is the angle

about the eigenvector that separates the two sets of coordinates. Such a set of four numbers is called a.quaternion While the quaternion as described above, does not involve

complex number s, if quaternions are used to describe two successive rotations, they must be combined using the non-commutativequaternion algebra derived byWilliam Rowan Hamilton through the use of imaginary numbers.Rotation calculation via quaternions has come to replace the use of

direction cosines in Aerospace applications through their reduction of the required calculations, and their ability to minimizeround-off error s. Also, incomputer graphics the ability to perform spherical interpolation between quaternions with relative ease is of value.**ee also***

Euler pole

*Euler angles

*Euler-Rodrigues parameters

* Rotation representation

*Rotation operator (vector space)

*Wikimedia Foundation.
2010.*

### Look at other dictionaries:

**Rotation representation (mathematics)**— In geometry a rotation representation expresses the orientation of an object (or coordinate frame) relative to a coordinate reference frame. This concept extends to classical mechanics where rotational (or angular) kinematics is the science of… … Wikipedia**Rotation group**— This article is about rotations in three dimensional Euclidean space. For rotations in four dimensional Euclidean space, see SO(4). For rotations in higher dimensions, see orthogonal group. In mechanics and geometry, the rotation group is the… … Wikipedia**Euler angles**— The Euler angles were developed by Leonhard Euler to describe the orientation of a rigid body (a body in which the relative position of all its points is constant) in 3 dimensional Euclidean space. To give an object a specific orientation it may… … Wikipedia**Rotation matrix**— In linear algebra, a rotation matrix is a matrix that is used to perform a rotation in Euclidean space. For example the matrix rotates points in the xy Cartesian plane counterclockwise through an angle θ about the origin of the Cartesian… … Wikipedia**Rotation operator (vector space)**— This article derives the main properties of rotations in 3 dimensional space.The three Euler rotations is an obvious way to bring a rigid body into any desired orientation bysequentially making rotations about axis fixed relative the body. But it … Wikipedia**Euler's three-body problem**— In physics and astronomy, Euler s three body problem is to solve for the motion of a particle that is acted upon by the gravitational field of two other point masses that are either fixed in space or move in circular coplanar orbits about their… … Wikipedia**Rotation system**— In combinatorial mathematics, rotation systems encode embeddings of graphs onto orientable surfaces, by describing the circular ordering of a graph s edges around each vertex.A more formal definition of a rotation system involves pairs of… … Wikipedia**List of topics named after Leonhard Euler**— In mathematics and physics, there are a large number of topics named in honour of Leonhard Euler (pronounced Oiler ). As well, many of these topics include their own unique function, equation, formula, identity, number (single or sequence), or… … Wikipedia**Conversion between quaternions and Euler angles**— Spatial rotations in three dimensions can be parametrized using both Euler angles and unit quaternions. This article explains how to convert between the two representations. Actually this simple use of quaternions was first presented by Euler… … Wikipedia**Quaternions and spatial rotation**— Unit quaternions provide a convenient mathematical notation for representing orientations and rotations of objects in three dimensions. Compared to Euler angles they are simpler to compose and avoid the problem of gimbal lock. Compared to… … Wikipedia