Versor
He was then able to display the general quaternion in polar coordinate form where T q is the norm of q.
The right versors form a sphere of square roots of −1 in the quaternion algebra.
Other versors include the twenty-four Hurwitz quaternions that have the norm 1 and form vertices of a 24-cell polychoron.
That's why it may be natural to understand corresponding versors as directed arcs that connect pairs of unit vectors and lie on a great circle formed by intersection of Π with the unit sphere, where the plane Π passes through the origin.
Arcs of the same direction and length (or, the same, subtended angle in radians) are equipollent and correspond to the same versor.
[1] Such an arc, although lying in the three-dimensional space, does not represent a path of a point rotating as described with the sandwiched product with the versor.
Indeed, it represents the left multiplication action of the versor on quaternions that preserves the plane Π and the corresponding great circle of 3-vectors.
Evidently versors are the image of the exponential map applied to a ball of radius π in the quaternion subspace of vectors.
Since quaternions can be interpreted as an algebra of two complex dimensions, the rotation action can also be viewed through the special unitary group SU(2).
[5] Versors have been used to represent rotations of the Bloch sphere with quaternion multiplication.
One of the methods of viewing elliptic space uses the Cayley transform to map the versors to
The set of all versors, with their multiplication as quaternions, forms a continuous group G. For a fixed pair
By a process of bitruncation of the 24-cell, the 48-cell on G is obtained, and these versors multiply as the binary octahedral group.
Another subgroup is formed by 120 icosians which multiply in the manner of the binary icosahedral group.
It is defined as a quantity of the form Such elements arise in split algebras, for example split-complex numbers or split-quaternions.
It was the algebra of tessarines discovered by James Cockle in 1848 that first provided hyperbolic versors.
In fact, Cockle wrote the above equation (with j in place of r) when he found that the tessarines included the new type of imaginary element.
[9][10] The primary exponent of hyperbolic versors was Alexander Macfarlane, as he worked to shape quaternion theory to serve physical science.
In a widely seen review, Macfarlane wrote: Today the concept of a one-parameter group subsumes the concepts of versor and hyperbolic versor as the terminology of Sophus Lie has replaced that of Hamilton and Macfarlane.
takes the real line to a group of hyperbolic or ordinary versors.
In the ordinary case, when r and −r are antipodes on a sphere, the one-parameter groups have the same points but are oppositely directed.
Robb (1911) defined the parameter rapidity, which specifies a change in frame of reference.
This rapidity parameter corresponds to the real variable in a one-parameter group of hyperbolic versors.
With the further development of special relativity the action of a hyperbolic versor came to be called a Lorentz boost.
The special orthogonal group SO(3,r) of rotations in three dimensions is closely related: it is a 2:1 homomorphic image of SU(2,c).
is just double the cross product of two vectors, which forms the multiplication operation in the Lie algebra.
The close relation to SU(1,c) and SO(3,r) is evident in the isomorphism of their Lie algebras.
The word is derived from Latin versari = "to turn" with the suffix -or forming a noun from the verb (i.e. versor = "the turner").
A rotation can be considered the result of two reflections, so it turns out a quaternion versor
In a departure from Hamilton's definition, multivector versors are not required to have unit norm, just to be invertible.
