In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a versor is a
quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...
of
norm
Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envir ...
one (a ''
unit
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of action, a discrete piece of action (or beat) in a theatrical presentation
Music
* ''Unit'' (alb ...
quaternion''). The word is derived from Latin ''versare'' = "to turn" with the suffix ''-or'' forming a noun from the verb (i.e. ''versor'' = "the turner"). It was introduced by
William Rowan Hamilton
Sir William Rowan Hamilton Doctor of Law, LL.D, Doctor of Civil Law, DCL, Royal Irish Academy, MRIA, Royal Astronomical Society#Fellow, FRAS (3/4 August 1805 – 2 September 1865) was an Irish mathematician, astronomer, and physicist. He was the ...
in the context of his quaternion theory.
Each versor has the form
:
where the r
2 = −1 condition means that r is a unit-length vector quaternion (or that the first component of r is zero, and the last three components of r are a
unit vector
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat").
The term ''direction vecto ...
in 3 dimensions). The corresponding
3-dimensional
Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informal ...
rotation has the angle 2''a'' about the axis r in
axis–angle representation
In mathematics, the axis–angle representation of a rotation parameterizes a rotation in a three-dimensional Euclidean space by two quantities: a unit vector indicating the direction of an axis of rotation, and an angle describing the magnitu ...
. In case (a
right angle
In geometry and trigonometry, a right angle is an angle of exactly 90 Degree (angle), degrees or radians corresponding to a quarter turn (geometry), turn. If a Line (mathematics)#Ray, ray is placed so that its endpoint is on a line and the ad ...
), then
, and the resulting unit vector is termed a ''
right versor
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat").
The term ''direction ve ...
''.
Presentation on 3- and 2-spheres
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 norm of ''q''. The norm of a versor is always equal to one; hence they occupy the unit
3-sphere
In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It may be embedded in 4-dimensional Euclidean space as the set of points equidistant from a fixed central point. Analogous to how the boundary of a ball in three dimensi ...
in H. Examples of versors include the eight elements of the
quaternion group
In group theory, the quaternion group Q8 (sometimes just denoted by Q) is a non-abelian group of order eight, isomorphic to the eight-element subset
\ of the quaternions under multiplication. It is given by the group presentation
:\mathrm_8 ...
. 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'' are examples of right versors, as well as their
additive inverse
In mathematics, the additive inverse of a number is the number that, when added to , yields zero. This number is also known as the opposite (number), sign change, and negation. For a real number, it reverses its sign: the additive inverse (opp ...
s. Other versors include the twenty-four
Hurwitz quaternion In mathematics, a Hurwitz quaternion (or Hurwitz integer) is a quaternion whose components are ''either'' all integers ''or'' all half-integers (halves of odd integers; a mixture of integers and half-integers is excluded). The set of all Hurwitz qu ...
s that have the norm 1 and form vertices of a
24-cell
In geometry, the 24-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also called C24, or the icositetrachoron, octaplex (short for "octahedral complex"), icosatetrahedroid, oct ...
polychoron.
Hamilton defined a quaternion as the quotient of two vectors. A versor can be defined as the quotient of two unit vectors. For any fixed
plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* ''Planes' ...
Π the quotient of two unit vectors lying in Π depends only on the
angle
In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''Side (plane geometry), sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle.
Angles formed by two ...
(directed) between them, the same ''a'' as in the unit vector–angle representation of a versor explained above. 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
In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point.
Any arc of a great circle is a geodesic of the sphere, so that great circles in spherical geomet ...
formed by intersection of Π with the
unit sphere
In mathematics, a unit sphere is simply a sphere of radius one around a given center. More generally, it is the set of points of distance 1 from a fixed central point, where different norms can be used as general notions of "distance". A unit b ...
, where the plane Π passes through the origin. Arcs of the same direction and length (or, the same,
its subtended angle in
radian
The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that c ...
s) are
equivalent
Equivalence or Equivalent may refer to:
Arts and entertainment
*Album-equivalent unit, a measurement unit in the music industry
* Equivalence class (music)
*'' Equivalent VIII'', or ''The Bricks'', a minimalist sculpture by Carl Andre
*''Equiva ...
, i.e. define the same versor.
Such an arc, although lying in the
three-dimensional space
Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position (geometry), position of an element (i.e., Point (m ...
, 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. The 3-dimensional rotation defined by the versor has the angle two times the arc's subtended angle, and preserves the same plane. It is a rotation about the corresponding vector r, that is
perpendicular
In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It can ...
to Π.
On three unit vectors, Hamilton writes
:
and
:
imply
:
Multiplication of quaternions of norm one corresponds to the (non-commutative) "addition" of great circle arcs on the unit sphere. Any pair of great circles either is the same circle or has two
intersection point
In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their ...
s. Hence, one can always move the point ''B'' and the corresponding vector to one of these points such that the beginning of the second arc will be the same as the end of the first arc.
An equation
:
implicitly specifies the unit vector–angle representation for the product of two versors. Its solution is an instance of the general
Campbell–Baker–Hausdorff formula in
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
theory. As the 3-sphere represented by versors in
is a 3-parameter Lie group, practice with versor compositions is a step into
Lie theory
In mathematics, the mathematician Sophus Lie ( ) initiated lines of study involving integration of differential equations, transformation groups, and contact of spheres that have come to be called Lie theory. For instance, the latter subject is L ...
. Evidently versors are the image of the
exponential map applied to a ball of radius π in the quaternion subspace of vectors.
Versors compose as aforementioned vector arcs, and Hamilton referred to this
group operation
In mathematics, a group is a set and an operation that combines any two elements of the set to produce a third element of the set, in such a way that the operation is associative, an identity element exists and every element has an inverse. Thes ...
as "the sum of arcs", but as quaternions they simply multiply.
The geometry of
elliptic space
Elliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel lines since any two lines must intersect. However, unlike in spherical geometry, two lines a ...
has been described as the space of versors.
Representation of SO(3)
The
orthogonal group
In mathematics, the orthogonal group in dimension , denoted , is the Group (mathematics), group of isometry, distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by ...
in three dimensions,
rotation group SO(3)
In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space \R^3 under the operation of composition.
By definition, a rotation about the origin is a tr ...
, is frequently interpreted with versors via the
inner automorphism
In abstract algebra an inner automorphism is an automorphism of a group, ring, or algebra given by the conjugation action of a fixed element, called the ''conjugating element''. They can be realized via simple operations from within the group itse ...
where ''u'' is a versor. Indeed, if
:
and vector ''s'' is perpendicular to ''r'',
then
:
by calculation. The plane
is isomorphic to
and the inner automorphism, by commutativity, reduces to the identity mapping there.
Since quaternions can be interpreted as an algebra of two complex dimensions, the rotation
action
Action may refer to:
* Action (narrative), a literary mode
* Action fiction, a type of genre fiction
* Action game, a genre of video game
Film
* Action film, a genre of film
* ''Action'' (1921 film), a film by John Ford
* ''Action'' (1980 fil ...
can also be viewed through the
special unitary group
In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1.
The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special ...
SU(2)
In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1.
The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special ...
.
For a fixed r, versors of the form exp(''a''r) where ''a'' ∈ , form a
subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
isomorphic to the
circle group
In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers.
\mathbb T = \ ...
. Orbits of the left multiplication action of this subgroup are fibers of a
fiber bundle
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...
over the 2-sphere, known as
Hopf fibration
In the mathematical field of differential topology, the Hopf fibration (also known as the Hopf bundle or Hopf map) describes a 3-sphere (a hypersphere in four-dimensional space) in terms of circles and an ordinary sphere. Discovered by Heinz Ho ...
in the case r = ''i''; other vectors give isomorphic, but not identical fibrations. In 2003 David W. Lyons wrote "the fibers of the Hopf map are circles in S
3" (page 95). Lyons gives an elementary introduction to quaternions to elucidate the Hopf fibration as a mapping on unit quaternions.
Versors have been used to represent rotations of the
Bloch sphere
In quantum quantum mechanics, mechanics and Quantum computing, computing, the Bloch sphere is a geometrical representation of the pure state space of a two-level system, two-level quantum mechanical system (qubit), named after the physicist Felix ...
with quaternion multiplication.
Elliptic space
The facility of versors illustrate
elliptic geometry
Elliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel lines since any two lines must intersect. However, unlike in spherical geometry, two lines a ...
, in particular
elliptic space
Elliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel lines since any two lines must intersect. However, unlike in spherical geometry, two lines a ...
, a three-dimensional realm of rotations. The versors are the points of this elliptic space, though they refer to
rotations in 4-dimensional Euclidean space
In mathematics, the group of rotations about a fixed point in four-dimensional Euclidean space is denoted SO(4). The name comes from the fact that it is the special orthogonal group of order 4.
In this article '' rotation'' means ''rotational ...
. Given two fixed versors ''u'' and ''v'', the mapping
is an ''elliptic motion''. If one of the fixed versors is 1, then the motion is a ''Clifford translation'' of the elliptic space, named after
William Kingdon Clifford
William Kingdon Clifford (4 May 18453 March 1879) was an English mathematician and philosopher. Building on the work of Hermann Grassmann, he introduced what is now termed geometric algebra, a special case of the Clifford algebra named in his ...
who was a proponent of the space. An elliptic line through versor ''u'' is
Parallelism in the space is expressed by
Clifford parallel In elliptic geometry, two lines are Clifford parallel or paratactic lines if the perpendicular distance between them is constant from point to point. The concept was first studied by William Kingdon Clifford in elliptic space and appears only in s ...
s. One of the methods of viewing elliptic space uses the
Cayley transform In mathematics, the Cayley transform, named after Arthur Cayley, is any of a cluster of related things. As originally described by , the Cayley transform is a mapping between skew-symmetric matrices and special orthogonal matrices. The transform ...
to map the versors to
Hyperbolic versor
A hyperbolic versor is a generalization of quaternionic versors to
indefinite orthogonal group
In mathematics, the indefinite orthogonal group, is the Lie group of all linear transformations of an ''n''-dimension of a vector space, dimensional real vector space that leave invariant a nondegenerate form, nondegenerate, symmetric bilinear for ...
s, such as
Lorentz group
In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physicis ...
.
It is defined as a quantity of the form
:
where
Such elements arise in algebras of
mixed signature, for example
split-complex number
In algebra, a split complex number (or hyperbolic number, also perplex number, double number) has two real number components and , and is written z=x+yj, where j^2=1. The ''conjugate'' of is z^*=x-yj. Since j^2=1, the product of a number wi ...
s or
split-quaternion
In abstract algebra, the split-quaternions or coquaternions form an algebraic structure introduced by James Cockle in 1849 under the latter name. They form an associative algebra of dimension four over the real numbers.
After introduction in th ...
s. It was the algebra of
tessarines
In abstract algebra, a bicomplex number is a pair of complex numbers constructed by the Cayley–Dickson process that defines the bicomplex conjugate (w,z)^* = (w, -z), and the product of two bicomplex numbers as
:(u,v)(w,z) = (u w - v z, u z ...
discovered by
James Cockle
Sir James Cockle FRS FRAS FCPS (14 January 1819 – 27 January 1895) was an English lawyer and mathematician.
Cockle was born on 14 January 1819. He was the second son of James Cockle, a surgeon, of Great Oakley, Essex. Educated at Charterho ...
in 1848 that first provided hyperbolic versors. In fact, James Cockle wrote the above equation (with in place of ) when he found that the tessarines included the new type of imaginary element.
This versor was used by
Homersham Cox (1882/83) in relation to quaternion multiplication. The primary exponent of hyperbolic versors was
Alexander Macfarlane
Alexander Macfarlane FRSE LLD (21 April 1851 – 28 August 1913) was a Scottish logician, physicist, and mathematician.
Life
Macfarlane was born in Blairgowrie, Scotland, to Daniel MacFarlane (Shoemaker, Blairgowire) and Ann Small. He s ...
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 quaternion
In abstract algebra, the algebra of hyperbolic quaternions is a nonassociative algebra over the real numbers with elements of the form
:q = a + bi + cj + dk, \quad a,b,c,d \in \mathbb \!
where the squares of i, j, and k are +1 and distinct eleme ...
s to extend the concept to 4-space. Problems in that algebra led to use of
biquaternion
In abstract algebra, the biquaternions are the numbers , where , and are complex numbers, or variants thereof, and the elements of multiply as in the quaternion group and commute with their coefficients. There are three types of biquaternions co ...
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.
Science
Science is a systematic endeavor that builds and organizes knowledge in the form of testable explanations and predictions about the universe.
Science may be as old as the human species, and some of the earliest archeological evidence for ...
, 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
Marius Sophus Lie ( ; ; 17 December 1842 – 18 February 1899) was a Norwegian mathematician. He largely created the theory of continuous symmetry and applied it to the study of geometry and differential equations.
Life and career
Marius Sophu ...
has replaced that of Hamilton and Macfarlane.
In particular, for each such that or , the mapping
takes the
real line
In elementary mathematics, a number line is a picture of a graduated straight line (geometry), line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real ...
to a group of hyperbolic or ordinary versors. In the ordinary case, when and − are
antipodes
In geography, the antipode () of any spot on Earth is the point on Earth's surface diametrically opposite to it. A pair of points ''antipodal'' () to each other are situated such that a straight line connecting the two would pass through Ear ...
on a sphere, the one-parameter groups have the same points but are oppositely directed. In physics, this aspect of
rotational symmetry
Rotational symmetry, also known as radial symmetry in geometry, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which i ...
is termed a
doublet
Doublet is a word derived from the Latin ''duplus'', "twofold, twice as much",