Question

I want to get a random 2D slice from a 3D volume in a computer program for generating noise. I decided to use a rotation matrix, which I understand has to be a member of the rotation group SO(3). How can I create a uniformly random member of this group?

Answer 1

Choose three random values u₁, u₂, and u₃ between 0 and 1.

A uniform random quaternion is given by:

• w = sqrt(1 – u₁)sin(2pi*u₂)

• x = sqrt(1 – u₁)cos(2pi*u₂)

• y = sqrt(u₁)sin(2pi*u₃)

• z = sqrt(u₁)cos(2pi*u₃)

You can then generate the appropriate rotation matrix using this formula:

    |       2     2                                |
    | 1 - 2y  - 2z    2xy - 2zw      2xz + 2yw     |
    |                                              |
    |                       2     2                |
M = | 2xy + 2zw       1 - 2x  - 2z   2yz - 2xw     |
    |                                              |
    |                                      2     2 |
    | 2xz - 2yw       2yz + 2xw      1 - 2x  - 2y  |
    |                                              |