Quaternions and Rotations

In "Best solution to an overdetermined system," TMJ 6(3): 20, a least squares method was used to compute numerically the quaternion that corresponds to the rotation matrix

through the equivalence of to the general (quaternion) rotation matrix:

P. Chesson (chessonp@aol.com) pointed out the quaternions can be determined directly. Since is a rotation matrix, its determinant

must be 1. This condition and the equation determines a Gršbner basis for the quaternion variables.

We can now use to simplify any combination of quaternions. For the monomials

the reduced polynomials are

and, since

the (arbitrary) quaternion is

For example, the numerical rotation matrix

with elements

corresponds to the quaternion

The positive square root for ensures represents a proper rotation. The equivalence of and is easily tested.