Euler's rotation theorem

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Euler's rotation theorem states that, in 3D space, for any two coordinate systems with a common origin, there is a single eigenvector with the same description in either system.

It follows that the angular relationship between the two systems may be expressed as a rotation about this Eigen axis, through some specific angle, u. In other words, 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 Eigen vector. The fourth is the angle about the Eigen vector that separates the two sets of coordinates. Such a set of four numbers is called a Quaternion.

The theorem is named after Leonhard Euler.

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.