Such a versor may be viewed as a directed arc of a circle with radius 1, representing the path of a point which rotates by an angle a about an axis r. In case a = π/2, the versor is a right versor. Left multiplication qz of a quaternion z to a versor q is identical to the action of the special unitary group SU(2) on the 2-dimensional complex space; hence, quaternionic versors are the traditional term and presentation for elements of SU(2). When used to represent a 3-dimensional rotation, a versor will rotate any quaternion vector v through the angle a around the unit vector r through the adjoint map
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 norm 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.
A sphere may be imagined to have two poles and an equator. If the sphere has radius 1, one of its poles can be represented by a unit vector with origin on its center O, and an arc of its equator with length a can be represented by the versor
The variable a can be viewed both as the length of an arc and a rotation angle. Namely, the above-mentioned arc can be viewed as the path of a point which rotates by a given angle about the center of the sphere O. Since the equator of the sphere is a great circle with radius 1, by definition the angle swept by that point, expressed in radians, is equal to the length of the arc along which the point moves.
Multiplication of quaternions of norm one corresponds to the "addition" of great circle arcs on the sphere. (See figure on right.) Hamilton writes
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.
In 1931 Heinz Hopf showed that the 3-sphere is a fiber bundle over the 2-sphere. In terms of versors q = exp(a r ), the fibers correspond to the parameter a. In 2003 David W. Lyons wrote "the fibers of the Hopf map are circles in S3" (page 95). Lyons gives an elementary introduction to quaternions to elucidate the Hopf fibration as a mapping on unit quaternions.
Such elements arise in algebras of mixed signature, 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, James Cockle wrote the above equation (with j in place of r) when he found that the tessarines included the new type of imaginary element.
The primary exponent of hyperbolic versors was Alexander Macfarlane as he worked to shape quaternion theory to serve physical science. He saw the modelling power of hyperbolic versors operating on the split-complex number plane, and in 1891 he introduced hyperbolic quaternions to extend the concept to 4-space. Problems in that algebra led to use of biquaternions after 1900. In a widely circulated review of 1899, Macfarlane said:
- …the root of a quadratic equation may be versor in nature or scalar in nature. If it is versor in nature, then the part affected by the radical involves the axis perpendicular to the plane of reference, and this is so, whether the radical involves the square root of minus one or not. In the former case the versor is circular, in the latter hyperbolic.
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. In particular, for each r such that r r = +1 or r r = −1, the mapping takes the real line to a group of hyperbolic or ordinary versors. In the ordinary case, when r and −r are antipodal points on a sphere, the one-parameter groups have the same points but are oppositely directed. In physics, this aspect of rotational symmetry is termed a doublet.
In 1911 Alfred Robb published his Optical Geometry of Motion in which he identified 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.
- Quaternions and spatial rotation
- Elliptic geometry #Elliptic space
- Rotations in 4-dimensional Euclidean space
- Elements of Quaternions, 2nd edition, v. 1, p. 146
- Lyons, David W. (April 2003), "An Elementary Introduction to the Hopf Fibration" (PDF), Mathematics Magazine 76 (2): 87–98, doi:10.2307/3219300, ISSN 0025-570X, JSTOR 3219300
- Papers on Space Analysis (1894), papers 2, 3, and 5, see external link below.
- Science, 9:326 (1899)
- W.R. Hamilton (1899) Elements of Quaternions, 2nd edition, edited by Charles Jasper Joly, Longmans Green & Company. See pp. 135–147.
- A.S. Hardy (1887) Elements of Quaternions, pp. 71,2 "Representation of Versors by spherical arcs" and pp. 112–8 "Applications to Spherical Trigonometry".
- C.C. Silva & R.A. Martins (2002) "Polar and Axial Vectors versus Quaternions", American Journal of Physics 70:958. Section IV: Versors and unitary vectors in the system of quaternions. Section V: Versor and unitary vectors in vector algebra.
- Pieter Molenbroeck (1891) Theorie der Quaternionen, Seite 48, "Darstellung der Versoren mittelst Bogen auf der Einheitskugel", Leiden: Brill.