- 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)

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