In geometry, 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 single rotation about some axis that runs through the fixed point. It also means that the composition of two rotations is also a rotation. Therefore the set of rotations has a structure known as a rotation group.
The theorem is named after Leonhard Euler, who proved it in 1775 by an elementary geometric argument. The axis of rotation is known as an Euler pole. The extension of the theorem to kinematics yields the concept of instant axis of rotation.
In linear algebra terms, the theorem states that, in 3D space, any two Cartesian 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 and that for a non-identity rotation matrix it must happen that: one of its eigenvalues is 1 and the other two are -1, or it has only one real eigenvalue which is equal to unity. The eigenvector corresponding to this eigenvalue is the axis of rotation connecting the two systems.
Read more about Euler's Rotation Theorem: Euler's Theorem (1776), Matrix Proof, Equivalence of An Orthogonal Matrix To A Rotation Matrix, Equivalence Classes, Generalizations
Famous quotes containing the words rotation and/or theorem:
“The lazy manage to keep up with the earth’s rotation just as well as the industrious.”
—Mason Cooley (b. 1927)
“To insure the adoration of a theorem for any length of time, faith is not enough, a police force is needed as well.”
—Albert Camus (1913–1960)