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. Generalizations In higher dimensions, any rigid motion that preserves a point in dimension or is a composition of at most rotations in orthogonal planes of rotation, though these planes need not be uniquely determined, and a rigid motion may fix multiple axes. A rigid motion in three dimensions that does not necessarily fix a point is a "screw motion". This is because a composition of a rotation with a translation perpendicular to the axis is a rotation about a parallel axis, while composition with a translation parallel to the axis yields a screw motion; see screw axis. This gives rise to screw theory. See also Euler angles Euler–Rodrigues parameters Rotation formalisms in three dimensions Angular velocity Rotation around a fixed axis Matrix exponential Axis–angle representation Chasles' theorem (kinematics), for an extension concerning general rigid body displacements. Notes References Euler's theorem and its proof are contained in paragraphs 24–26 of the appendix (Additamentum. pp. 201–203) of L. Eulero (Leonhard Euler), Formulae generales pro translatione quacunque corporum rigidorum (General formulas for the translation of arbitrary rigid bodies), presented to the St. Petersburg Academy on October 9, 1775, and first published in Novi Commentarii academiae scientiarum Petropolitanae 20, 1776, pp. 189–207 (E478) and was reprinted in Theoria motus corporum rigidorum, ed. nova, 1790, pp. 449–460 (E478a) and later in his collected works Opera Omnia, Series 2, Volume 9, pp. 84–98. External links Euler's original treatise in The Euler Archive: entry on E478, first publication 1776 (pdf) Euler's original text (in Latin) and English translation (by Johan Sten) Wolfram Demonstrations Project for Euler's Rotation Theorem (by Tom Verhoeff) Euclidean symmetries Theorems in geometry Rotation in three dimensions Leonhard Euler