Euler's rotation theorem

In <a href="/facts/Geometry/wTQf5ffQ">geometry</a>, Euler's rotation theorem states that, in <a href="/facts/Three-dimensional_space/KAqKWUbS">three-dimensional space</a>, any displacement of a <a href="/facts/Rigid_body/THezBlq2">rigid body</a> such that a point on the rigid body remains fixed, is equivalent to a single rotation about some axis that runs through the <a href="/facts/Fixed_point_(mathematics)/vVVddTKc">fixed point</a>. It also means that the composition of two rotations is also a rotation. Therefore the set of rotations has a group structure, known as a <a href="/facts/3D_rotation_group/yXbKPzhT">rotation group</a>.
The theorem is named after <a href="/facts/Leonhard_Euler/z2fsbjgj">Leonhard Euler</a>, who proved it in 1775 by means of <a href="/facts/Spherical_geometry/WGLQb1Ys">spherical geometry</a>. The axis of rotation is known as an Euler axis, typically represented by a <a href="/facts/Unit_vector/GyaXneDn">unit vector</a> ê. Its product by the rotation angle is known as an <a href="/facts/Axis-angle/WDfmToVP">axis-angle vector</a>. The extension of the theorem to <a href="/facts/Kinematics/o7mSyeJA">kinematics</a> yields the concept of <a href="/facts/Instant_axis_of_rotation/x6JgAtTH">instant axis of rotation</a>, a line of fixed points.
In linear algebra terms, the theorem states that, in 3D space, any two <a href="/facts/Cartesian_coordinate_systems/lkTqvMmB">Cartesian coordinate systems</a> 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 <a href="/facts/Rotation_matrix/c2K2Ftl8">rotation matrix</a> one <a href="/facts/Eigenvalue/8TjEoT8u">eigenvalue</a> is 1 and the other two are both complex, or both equal to −1. The <a href="/facts/Eigenvector/8TjEoT8u">eigenvector</a> corresponding to this eigenvalue is the axis of rotation connecting the two systems.

Euler's rotation theorem open-in-new

Euler's rotation theorem