## Quaternions and RotationsIn "Best solution to an overdetermined system," 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. |