Editing 3D rotation group (section)

== Uniform random sampling ==
<math>SO(3)</math> is doubly covered by the group of unit quaternions, which is isomorphic to the 3-sphere. Since the [[Haar measure]] on the unit quaternions is just the 3-area measure in 4 dimensions, the Haar measure on <math>SO(3)</math> is just the pushforward of the 3-area measure.

Consequently, generating a uniformly random rotation in <math>\R^3</math> is equivalent to generating a uniformly random point on the 3-sphere. This can be accomplished by the following<math display="block">(\sqrt{1-u_1}\sin(2\pi u_2), \sqrt{1-u_1}\cos(2\pi u_2), \sqrt{u_1}\sin(2\pi u_3), \sqrt{u_1}\cos(2\pi u_3))</math>

where <math>u_1, u_2, u_3</math> are uniformly random samples of <math>[0, 1]</math>.<ref>{{Citation |last=Shoemake |first=Ken |title=III.6 - Uniform Random Rotations |date=1992-01-01 |url=https://www.sciencedirect.com/science/article/pii/B9780080507552500361 |work=Graphics Gems III (IBM Version) |pages=124–132 |editor-last=Kirk |editor-first=DAVID |place=San Francisco |publisher=Morgan Kaufmann |language=en |isbn=978-0-12-409673-8 |access-date=2022-07-29}}</ref>