In
mathematicsMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
, a
versor is a directed great-circle arc that corresponds to a
quaternionIn mathematics, quaternions are a noncommutative number system that extends the complex numbers. The quaternions were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space...
of
normIn linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector...
one. In
geometryGeometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
and
physicsPhysics is a natural science; it is the study of matter and its motion through spacetime and all that derives from these, such as energy and force...
, a
versor is sometimes defined as a
unit vectorIn mathematics, a unit vector in a normed vector space is a vector whose length is 1 . A unit vector is often denoted by a lowercase letter with a superscribed caret or “hat”, like this: .In Euclidean space, the dot product of two unit vectors is simply the...
indicating the orientation of a directed axis (such as a
Cartesian axisA Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length....
) or of another vector.
The word is from Latin
versus = "turned", from pp. of
vertere = "to turn", and was introduced by
William Rowan HamiltonSir William Rowan Hamilton was an Irish physicist, astronomer, and mathematician, who made important contributions to classical mechanics, optics, and algebra. His studies of mechanical and optical systems led him to discover new mathematical concepts and techniques...
, in the context of his quaternion theory.
Hamilton denoted the
versor of a quaternion
q by the symbol
Uq.
In
mathematicsMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
, a
versor is a directed great-circle arc that corresponds to a
quaternionIn mathematics, quaternions are a noncommutative number system that extends the complex numbers. The quaternions were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space...
of
normIn linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector...
one. In
geometryGeometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
and
physicsPhysics is a natural science; it is the study of matter and its motion through spacetime and all that derives from these, such as energy and force...
, a
versor is sometimes defined as a
unit vectorIn mathematics, a unit vector in a normed vector space is a vector whose length is 1 . A unit vector is often denoted by a lowercase letter with a superscribed caret or “hat”, like this: .In Euclidean space, the dot product of two unit vectors is simply the...
indicating the orientation of a directed axis (such as a
Cartesian axisA Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length....
) or of another vector.
The word is from Latin
versus = "turned", from pp. of
vertere = "to turn", and was introduced by
William Rowan HamiltonSir William Rowan Hamilton was an Irish physicist, astronomer, and mathematician, who made important contributions to classical mechanics, optics, and algebra. His studies of mechanical and optical systems led him to discover new mathematical concepts and techniques...
, in the context of his quaternion theory.
Definition in quaternion theory
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 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 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
When versors are used for
spherical trigonometrySpherical trigonometry is a branch of spherical geometry, which deals with polygons on the sphere and the relationships between the sides and the angles...
we have an illustration of quaternion algebra in practical expression.
Since versors correspond to elements of the
3-sphereIn mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It consists of the set of points equidistant from a fixed central point in 4-dimensional Euclidean space...
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 groupIn mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
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 theoryLie theory is an area of mathematics, developed initially by Sophus Lie.In Lie's early work, the idea was to construct a theory of continuous groups, to complement the theory of discrete groups that had developed in the theory of modular forms, in the hands of Felix Klein and Henri Poincaré...
, the pair (group,algebra) carries this logarithm-likeness to higher dimensions.
Definition in geometry and physics
Since right versors play an important role in the description of
three-dimensional spaceThree-dimensional space is a geometric model of the physical universe in which we live. The three dimensions are commonly called length, width, and depth , although any three mutually perpendicular directions can serve as the three dimensions....
, the term
versor is frequently adopted to mean
right versor when no other quaternion terminology is used.
Thus 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
A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length....
are the unit vectors codirectional with the axes of that system.
- The versor (or normalized vector) of a non-zero vector is the unit vector codirectional with , i.e.,
-
- where is the norm
In linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector...
(or length) of .
Hyperbolic versor
In
linear algebraLinear algebra is a branch of mathematics concerned with the study of vectors, vector spaces , linear maps , and systems of linear equations. Vector spaces are a central theme in modern mathematics; thus, linear algebra is widely used in both abstract algebra and functional analysis...
, a hyperbolic versor is a quantity of the form.
Such elements arise in algebras of
mixed signatureThe signature of a metric tensor is the number of positive and negative eigenvalues of the metric. That is, the corresponding real symmetric matrix is diagonalised, and the diagonal entries of each sign counted...
, for example
split-complex numberIn abstract algebra, the split-complex numbers are a two-dimensional commutative algebra over the real numbers different from the complex numbers. Every split-complex number has the formwhere x and y are real numbers...
s or split-quaternions. It was the algebra of tessarines discovered by
James CockleSir James Cockle FRS FRAS FCPS FMS was an English lawyer andmathematician.Cockle was born on the 14th of January 1819. He was the second son of James Cockle, a surgeon, of Great Oakley, Essex...
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 MacfarlaneAlexander Macfarlane FRSE was a Scottish logician, physicist, and mathematician.Macfarlane was born in Blairgowrie, Scotland and studied at the University of Edinburgh. His doctoral thesis, on the conditions governing the electric spark, was subsequently published in the Transactions of the Royal...
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 quaternionIn mathematics, a hyperbolic quaternion is a mathematical concept first suggested by Alexander Macfarlane in 1891 in a speech to the American Association for the Advancement of Science. The idea was criticized for its failure to conform to associativity of multiplication, so the collection M of all...
s to extend the concept to 4-space. Problems in that algebra lead to use of
biquaternionThe biquaternions are the numbers where w, x, y, and z are complex numbers and the elements of {1, i, j, k} multiply as in the quaternion group...
s 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 groupIn mathematics, a one-parameter group or one-parameter subgroup usually means a continuous group homomorphismfrom the real line R to some other topological group G. That means that it is not in fact a group, strictly speaking; if φ is injective then φ, the image, will be a subgroup of G that is...
subsumes the concepts of versor and hyperbolic versor as the terminology of
Sophus LieMarius Sophus Lie was a Norwegian-born mathematician. He largely created the theory of continuous symmetry, and applied it to the study of geometry and differential equations.- Biography :...
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 pointIn mathematics, the antipodal point of a point on the surface of a sphere is the point which is diametrically opposite it — so situated that a line drawn from the one to the other passes through the centre of the sphere and forms a true diameter....
s on a sphere, the one-parameter groups have the same points but are oppositely directed. In physics, this aspect of
rotational symmetryGenerally speaking, an object with rotational symmetry is an object that looks the same after a certain amount of rotation. An object may have more than one rotational symmetry; for instance, if reflections or turning it over are not counted, the triskelion appearing on the Isle of Man's flag has...
is termed a
doublet (physics)In quantum mechanics, a doublet is a quantum state of a system with a spin of 1/2, such that there are two allowed values of the spin component, −1/2 and +1/2. Quantum systems with two possible states are sometimes called two-level systems...
.
In 1911
Alfred RobbAlfred Arthur Robb or Alfred A. Robb was an English physicist.Robb studied at Queen's College in Belfast and at St John’s College in Cambridge. He then proceeded to University of Göttingen, where guided by Woldemar Voigt, he wrote his dissertation on the Zeeman effect. He also worked under J. J...
published his
Optical Geometry of Motion in which he identified the parameter
rapidityIn relativity rapidity is an alternative to speed as a method of measuring motion. For low speeds, rapidity and speed are proportional, but for high speeds, rapidity takes a larger value. The rapidity of light is infinite....
which specifies a change in
frame of referenceA frame of reference in physics, may refer to a coordinate system or set of axes within which to measure the position, orientation, and other properties of objects in it, or it may refer to an observational reference frame tied to the state of motion of an observer.It may also refer to both an...
. This rapidity parameter corresponds to the real variable in a one-parameter group of hyperbolic versors. With the further development of
special relativitySpecial relativity is the physical theory of measurement in inertial frames of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies"...
the action of a hyperbolic versor came to be called a Lorentz boost.
External links
- http://www.biology-online.org/dictionary/versor
- http://www.thefreedictionary.com/Versor