# Sampling a Uniformly Random Rotation

Sampling a Uniformly Random Rotation

The orientation of a sphere is an element of and can be represented by three Euler angles. However, uniformly sampling three Euler angles does not result in a uniform sampling of . To generate a uniformly distributed random rotation in , first perform a random rotation about the axis, then rotate the axis to a random position on the sphere.

SO(3)

SO(3)

SO(3)

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z

We can use this same method to generate uniformly distributed perturbations in by scaling the axis rotation and deflection by a number less than 1.

SO(3)

z