Editing Euler's rotation theorem (section)

{{Short description|Movement with a fixed point is rotation}}

[[Image:Euler AxisAngle.png|thumb|right|A rotation represented by an Euler axis and angle.]]

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 (mathematics)|fixed point]]. 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 ''[[3D rotation group|rotation group]]''.

The theorem is named after [[Leonhard Euler]], who proved it in 1775 by means of [[spherical geometry]]. The axis of rotation is known as an '''Euler axis''', typically represented by a [[unit vector]] '''ê'''. Its product by the rotation angle is known as an [[axis-angle|axis-angle vector]]. The extension of the theorem to [[kinematics]] yields the concept of [[instant axis of rotation]], a line of fixed points.

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]] one [[eigenvalue]] is 1 and the other two are both complex, or both equal to −1. The [[eigenvector]] corresponding to this eigenvalue is the axis of rotation connecting the two systems.