Euler's rotation theorem
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In kinematics, 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 rotation about a fixed axis through that point. The theorem is named after Leonhard 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 rotation matrix has one and only one real eigenvalue, which is equal to unity. The eigenvector corresponding to this eigenvalue is the axis of rotation connecting the two systems.
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[edit] Applications
[edit] 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:
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:
It can be seen that Euler's theorem essentially states that all rotations may be represented in this form. The product 
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 the Lie algebra of the rotation group.
[edit] 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 numbers, if quaternions are used to describe two successive rotations, they must be combined using the non-commutative quaternion algebra derived by William 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 minimize round-off errors. Also, in computer graphics the ability to perform spherical interpolation between quaternions with relative ease is of value.
