Improved Quaternion Rotation Formulas

  One can use a vector quaternion to represent the pole of a rotation. To do the rotation of any vector quaternion properly one has to one has to partition the quaternion into polar and normal parts, v_p and v_n. We need a third vector in the plane of rotation, the binormal, v_b, to do the rotation.

Another formula for rotating quaternions was given by Hamilton and is explained in Kellog and Tait's Introduction to Quaternions. Since the square of the magnitude of q̂ equals 1, we can replace q̂^-1 by its conjugate q̂*. We start by setting q̂=1cosφ+p̂ sinφ. The product of the three quaternions gives essentially the same formula for the rotation but with θ replaced by 2φ.