Versor - Presentation On The Sphere

Presentation On The Sphere

Hamilton denoted the versor of a quaternion q by the symbol Uq. He was then able to display the general quaternion in polar coordinate form

q = Tq Uq,

where Tq is the norm of q. The tensor of a versor is always equal to one. Of particular importance are the right versors, which have angle π/2. These versors have zero scalar part, and so are vectors of length one (unit vectors). The right versors form a sphere of square roots of −1 in the quaternion algebra. The generators i, j, and k in the quaternion group are examples of right versors.

If a great-circle arc has length a, and if is the pole of this great circle (viewed as the equator with respect to the pole), then the versor is the quaternion

Multiplication of quaternions of norm one corresponds to the "addition" of great circle arcs on the 2-sphere. Hamilton writes

and

imply

The algebra of versors has been exploited to exhibit the properties of elliptic space.

Since versors correspond to elements of the 3-sphere in H, it is natural today to write

for the versor composition, where is the pole of the product versor and b is its angle (as in the figure).

When we view the spherical trigonometric solution for b and in the product of exponentials, then we have an instance of the general Campbell-Baker-Hausdorff formula in Lie group theory. As the 3-sphere represented by versors in H is a 3-parameter Lie group, practice with versor compositions is good preparation for more abstract Lie group and Lie algebra theory. Indeed, as great circle arcs they compose as sums of vector arcs (Hamilton's term), but as quaternions they simply multiply. Thus the great-circle-arc model is similar to logarithm in that sums correspond to products. In Lie theory, the pair (group,algebra) carries this logarithm-likeness to higher dimensions.