Rotation in three dimensions

three angles introduced by Leonhard Euler to describe the orientation of a rigid body with respect to a fixed coordinate system

thumb|upright=1.5|A spinor visualized as a vector pointing along the Möbius band, exhibiting a sign inversion when the circle (the "physical system") is continuously rotated through a full turn of 360°.

quantum number parameterizing spin and angular momentum

coefficients in angular momentum eigenstates of quantum systems

description of the rotation of an item relative to defined coordinate axes of the space it occupies

quasilinear first-order ordinary differential equation describing the rotation of a rigid body

group of rotations in 3 dimensions

quantum number describing the total angular momentum of an atom

theorem

rotation in the plane spanned by two coordinates axes

thumb|A single point in space can spin continuously without becoming tangled. Notice that after a 360° rotation, the spiral flips between clockwise and counterclockwise orientations. It returns to its original configuration after spinning a full 720°.

correspondence between quaternions and 3D rotations

In mathematics, a versor is a quaternion whose norm is one, also known as a unit quaternion. Each versor has the form \ u = \exp(a\mathbf{r}) = \cos a + \mathbf{r} \sin a, \qquad \mathbf{r}^2 = -1, \qquad a \in [0,\pi]\ , where the condition \ \mathbf{r}^2 = -1\ means that \ \mathbf{r}\ is an algebraic imaginary unit. There is a sphere of imaginary units in the quaternions. Note that the expression for a versor is just Euler's formula for the imaginary unit \ \mathbf{r} ~. If \ a = \tfrac{\pi}{2}\ (when \ a\ is a right angle), then \ u = \mathbf{r}\ , and it is called a right versor.

vector formula for a rotation in space, given its axis

thumb|Animation of free (or Euler) nutation of a sphere thumb|upright=0.6| in obliquity of a planet caused by the Gravity|gravitational forces of other nearby bodies acting upon the planet

about translation of rigid bodies

Math formula for 3D vector rotation

first case of a Lie group that is both compact and non-abelian

ways to represent 3D rotations

method of quaternion rotation

parameterization of a rotation into a unit vector and angle