Versor


Versor

In mathematics, a versor is a directed great-circle arc that corresponds to a quaternion of norm one. In geometry and physics, a versor is sometimes defined as a unit vector indicating the orientation of a directed axis (such as a Cartesian axis) or of another vector.

The word is from Latin "versus" = "turned", from pp. of "vertere" = "to turn", and was introduced by William Rowan Hamilton, in the context of his quaternion theory.

Definition in quaternion theory

Hamilton denoted the versor of a quaternion "q" by the symbol U"q". He was then able to display the general quaternion in polar coordinate form: "q" = T"q" U"q", where T"q" is the “tensor 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). In all, H has a 2-sphere of right versors, of which i, j, and k are examples.

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

: exp(amathbf{r}) = cos a + mathbf{r} sin a. ,

Multiplication of quaternions of norm one corresponds to the “addition” of great circle arcs on the 2-sphere. Hamilton writes ["Elements of Quaternions", 2nd edition,v.1,p.146]

: q = eta: alpha = OB:OA and: q' = gamma:eta = OC:OB ,

imply

: q' q = gamma:alpha = OC:OA, ,.

When versors are used for spherical trigonometry we have an illustration of quaternion algebra in practical expression.

Quaternions in Lie Group Theory

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

: exp(cmathbf{r}) exp(amathbf{s}) = exp(bmathbf{t}) !

for the versor composition, where mathbf{t}, 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 mathbf{t} 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 compositons 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.

Definition in geometry and physics

A versor is sometimes defined as a unit vector indicating the direction of a directed axis or vector. For instance:

:* The versors of a Cartesian coordinate system are the unit vectors codirectional with the axes of that system.

:* The versor (or normalized vector) oldsymbol{hat{u of a non-zero vector oldsymbol{u} is the unit vector codirectional with oldsymbol{u}, i.e.,

:::oldsymbol{hat{u = frac{oldsymbol{u{|oldsymbol{u}.

::where |oldsymbol{u}| is the norm (or length) of oldsymbol{u}.

Hyperbolic versor

In linear algebra, a hyperbolic versor is a quantity of the form:exp(ar) = cosh a + r sinh a ext{ where } r^2 = +1.Such elements arise in algebras of mixed signature. Examples include split-complex numbers, split-quaternions, and biquaternions. 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 "r = j") 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 mould quaternion theory to serve physical science. He saw the modelling power of hyperbolic versors operating on the split-complex number plane, and he developed hyperbolic quaternions to extend the concept to 4-space. Problems in that algebra lead to use of biquaternions instead. In a widely circulated review, 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. [Science (journal), 9:326 (1899)] 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 a o exp(ar) 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 (physics).

ee also

*Quaternions and spatial rotation

References

* 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.

External links

* http://www.biology-online.org/dictionary/versor
* http://www.thefreedictionary.com/Versor


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