Sampling a Uniformly Random Rotation

number of sampled rotations

100

generate random

rotations

perturbations

z

axis rotation

z

axis deflection

The orientation of a sphere is an element of

SO(3)

and can be represented by three Euler angles. However, uniformly sampling three Euler angles does not result in a uniform sampling of

SO(3)

. To generate a uniformly distributed random rotation in

SO(3)

, first perform a random rotation about the

z

axis, then rotate the

z

axis to a random position on the sphere.

We can use this same method to generate uniformly distributed perturbations in

SO(3)

by scaling the

z

axis rotation and deflection by a number less than 1.