User:Prof McCarthy/Rodrigues equation

Quaternions and spatial rotations

A benefit of the quaternion formulation of the composition of two rotations R_B and R_A is that the resulting quaternion yields directly the rotation axis and angle of the composite rotation R_C=R_BR_A.

In general, the quaternion associated with a spatial rotation R is constructed from its rotation axis S and the rotation angle φ this axis. The associated quaternion is given by,

$S=\cos {\frac {\phi }{2}}+\sin {\frac {\phi }{2}}\mathbf {S} .$

Let the composition of the rotation R_R with R_A be the rotation R_C=R_BR_A. The rotation axis and angle of R_C is obtained from the product of the quaternions

$A=\cos(\alpha /2)+\sin(\alpha /2)\mathbf {A} \quad {\text{and}}\quad B=\cos(\beta /2)+\sin(\beta /2)\mathbf {B} .$

That is, the composite rotation R_C=R_BR_A is defined by the quaternion

$C=\cos {\frac {\gamma }{2}}+\sin {\frac {\gamma }{2}}\mathbf {C} ={\Big (}\cos {\frac {\beta }{2}}+\sin {\frac {\beta }{2}}\mathbf {B} {\Big )}{\Big (}\cos {\frac {\alpha }{2}}+\sin {\frac {\alpha }{2}}\mathbf {A} {\Big )}.$

Expand this product to obtain

$\cos {\frac {\gamma }{2}}+\sin {\frac {\gamma }{2}}\mathbf {C} ={\Big (}\cos {\frac {\beta }{2}}\cos {\frac {\alpha }{2}}-\sin {\frac {\beta }{2}}\sin {\frac {\alpha }{2}}\mathbf {B} \cdot \mathbf {A} {\Big )}+{\Big (}\sin {\frac {\beta }{2}}\cos {\frac {\alpha }{2}}\mathbf {B} +\sin {\frac {\alpha }{2}}\cos {\frac {\beta }{2}}\mathbf {A} +\sin {\frac {\beta }{2}}\sin {\frac {\alpha }{2}}\mathbf {B} \times \mathbf {A} {\Big )}.$

Divide both sides of this equation by the identity, which is the law of cosines on a sphere,

$\cos {\frac {\gamma }{2}}=\cos {\frac {\beta }{2}}\cos {\frac {\alpha }{2}}-\sin {\frac {\beta }{2}}\sin {\frac {\alpha }{2}}\mathbf {B} \cdot \mathbf {A} ,$

and compute

$\tan {\frac {\gamma }{2}}\mathbf {C} ={\frac {\tan {\frac {\beta }{2}}\mathbf {B} +\tan {\frac {\alpha }{2}}\mathbf {A} +\tan {\frac {\beta }{2}}\tan {\frac {\alpha }{2}}\mathbf {B} \times \mathbf {A} }{1-\tan {\frac {\beta }{2}}\tan {\frac {\alpha }{2}}\mathbf {B} \cdot \mathbf {A} }}.$

This is Rodrigues formula for the axis of a composite rotation defined in terms of the axes of the two rotations. He derived this formula in 1840 (see page 408).

The three rotation axes A, B, and C form a spherical triangle] and the dihedral angles at the vertices of the triangle between the planes formed by the sides of the this triangle are defined by the these rotation angles.