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| {{about|the quaternionic representation of a rotation}}
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| '''Versors''' are an [[algebra]]ic parametrisation of [[rotation (mathematics)|rotations]]. In [[classical Hamiltonian quaternions|classical quaternion theory]] a '''versor''' is a [[quaternion]] of [[norm (mathematics)|norm]] one (a ''unit quaternion'').
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| Each versor has the form
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| :<math>q = \exp(a\mathbf{r}) = \cos a + \mathbf{r} \sin a, \quad \mathbf{r}^2 = -1, \quad a \in [0,\pi],</math>
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| <!--
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| -->where the '''r'''<sup>2</sup> = −1 condition means that '''r''' is a 3-dimensional [[unit vector]]. In case {{nowrap|1=''a'' = π/2}}, the versor is termed a '''right versor'''.
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| The corresponding [[three-dimensional space|3-dimensional]] rotation has the angle {{num|2}}''a'' about the axis '''r''' in [[axis–angle representation]].
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| The word is from Latin ''[[wikt:versus|versus]]'' = "turned", from pp. of ''vertere'' = "to turn", and was introduced by [[William Rowan Hamilton]] in the context of his quaternion theory. Due to historic reasons, it sometimes is used synonymously with a "[[unit vector|unit]] quaternion" without a reference to rotations.<!-- should it link to [[versor (physics)]]? -->
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| ==Versors, rotations, and Lie groups==<!-- caution: internal #-links -->
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| In the quaternion algebra a versor <math>q = \exp(a \mathbf{r})</math> will rotate any quaternion ''v'' through the sandwiching product map <math> v \mapsto q v q^{-1} </math> such that the [[Quaternion #Scalar and vector parts|scalar part]] of ''v'' conserves. If this scalar part (the [[four-dimensional space|fourth dimension]] of the quaternion space) is zero, i.e. ''v'' is a [[Euclidean vector]] in three dimensions, then the formula above defines the rotation through [[axis–angle representation|the angle 2''a'' around the unit vector '''r''']]. For this case, this formula expresses the [[adjoint representation]] of the [[Spin group|Spin]](3) [[Lie group]] in its respective [[Lie algebra]] of 3-dimensional Euclidean vectors, and the factor "2" is due to the [[double covering group|double covering]] of Spin(3) over the [[rotation group SO(3)]]. In other words, {{gaps|''q''|''v''|''q''<sup>−1</sup>}} rotates the [[Quaternion #Scalar and vector parts|vector part]] of ''v'' around the vector '''r'''. See [[quaternions and spatial rotation]] for details.
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| {{anchor|left multiplication}}A quaternionic versor expressed in the [[Quaternion #Matrix representations|complex {{gaps|2|×|2}} matrix representation]] is an element of the [[special unitary group]] SU(2). Spin(3) and SU(2) are the same [[group (mathematics)|group]]. Left multiplication ''qv'' of a quaternion ''v'' to a versor ''q'' is another kind of quaternion rotation as a 4-dimensional real [[vector space]], identical to the SU(2) [[group action|action]] on the 2-dimensional [[complex number|complex]] space identical to quaternions (''v'' = ''A'' + ''Bj''). Angles of rotation in this [[weight (representation theory)|''λ'' = 1/2 representation]] are equal to ''a''; there is no "2" factor in angles unlike the ''λ'' = 1 adjoint representation mentioned [[#Versors, rotations, and Lie groups|above]]; see [[representation theory of SU(2)]] for details.
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| For a fixed '''r''', versors of the form exp(''a'''''r''') where ''a'' ∈ {{open-closed|−π, π}}, form a [[subgroup]] isomorphic to the [[circle group]]. Orbits of the left multiplication action of this subgroup are fibers of a [[fiber bundle]] over the 2-sphere, known as [[Hopf fibration]] in the case '''r''' = ''i''; other vectors give isomorphic, but not identical fibrations. In 2003 David W. Lyons<ref>{{citation | |
| | doi=10.2307/3219300
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| | last=Lyons
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| | first=David W.
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| | title=An Elementary Introduction to the Hopf Fibration
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| | journal=[[Mathematics Magazine]]
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| | volume=76
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| | issue=2
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| | pages=87–98
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| |date=April 2003
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| | url=http://csunix1.lvc.edu/~lyons/pubs/hopf_paper_preprint.pdf
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| | format =[[PDF]]
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| | issn=0025-570X
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| | jstor=3219300
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| }}</ref> wrote "the fibers of the Hopf map are circles in S<sup>3</sup>" (page 95). Lyons gives an elementary introduction to quaternions to elucidate the Hopf fibration as a mapping on unit quaternions.
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| ==Presentation on 3- and 2-spheres==
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| {{refimprove|section|date=January 2014}}
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| Hamilton denoted the '''versor''' of a quaternion ''q'' by the symbol '''U'''''q''. He was then able to display the general quaternion in [[polar decomposition#quaternion polar decomposition|polar coordinate form]]
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| : ''q'' = '''T'''''q'' '''U'''''q'',
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| 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 '''H'''. Examples of versors are 8 elements of the [[quaternion group]] and, more generally, 24 [[Hurwitz quaternion]]s that have the norm 1 and form vertices of a [[24-cell]] polychoron.
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| 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 (geometry)|plane]] Π the quotient of two unit vectors lying in Π depends only on the [[angle]] (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 [[arc (geometry)|arcs]] <!--(or [[line segment]]s with respect to the [[spherical geometry]])--> 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, [[arc (geometry)#Length of an arc of a circle|its subtended angle]] in [[radian]]s) are [[equivalence relation|equivalent]], i.e. define the same versor.
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| [[Image:Spherical triangle.svg|thumb|right|arc ''AB'' + arc ''BC'' = arc ''AC'']]
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| Such an arc, although lying in the [[three-dimensional space]], does not represent a path of a point rotating as [[#Versors, rotations, and Lie groups|described with the sandwiched product]] with the versor. Indeed, it represents the [[#left multiplication|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]] to Π.
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| [[Image:Spin 180,60.svg|thumb|right|300px|Two versors from orthogonal planes (such as {{math|''i''}} and {{sfrac|1|2}}{{math|({{sqrt|3}} + ''j'')}}) multiply in both orders, giving ''different'' products represented with different [[spherical triangle]]s (green)]]
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| On ''three'' unit vectors, Hamilton writes<ref>''Elements of Quaternions'', 2nd edition, v. 1, p. 146</ref>
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| : <math>q = \beta: \alpha = OB:OA \ </math> and
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| : <math>q' = \gamma:\beta = OC:OB </math>
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| imply
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| : <math> q' q = \gamma:\alpha = OC:OA . </math>
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| 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]]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 shall be the same as the end of the first arc.
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| Of particular importance{{which|date=January 2014}} are the [[Classical_Hamiltonian_quaternions#Right_versor|right versors]], which have [[right angle|angle π/2]]. These versors have zero scalar part, and so are [[Euclidean vector|vectors]] of length one (unit vectors). The right versors form a [[quaternion#Square roots of −1|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]]s.<!-- Only a Hurwitz quaternion that belongs to the quaternion group can be a right versor (follows from the definition in coordinates) -->
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| The algebra of versors has been exploited{{how|date=January 2014}} to exhibit the properties of [[elliptic space]].
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| An equation
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| : <math>\exp(c\mathbf{r}) \exp(a\mathbf{s}) = \exp(b\mathbf{t}) \!</math>
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| 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]] theory.{{fact|date=January 2014}} 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. They compose as aforementioned vector arcs, and Hamilton referred to this [[group (mathematics)|group operation]] as "the sum of arcs", but as quaternions they simply multiply. The vector ''a'''''r''' specifies the [[logarithm]] map from {{nowrap|Spin(3) ∖ {−1}<nowiki/>}} to an [[ball (mathematics)|open ball]] in the [[Euclidean space]], an inverse of the [[exponential map]]. Namely, sums of [[collinear]] vectors correspond to products of versors, and logarithm of the composition of rotations by two "small" versors ''approximately'' equals to the sum of their logarithms. In [[Lie theory]], the pair ({{gaps|group,|algebra}}) generalizes this logarithm map to higher (but not infinite) dimensions.
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| ==Hyperbolic versor==
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| A hyperbolic versor is a generalization of quaternionic versors to [[indefinite orthogonal group]]s, such as [[Lorentz group]].
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| It is defined as a quantity of the form
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| :<math>\exp(ar) = \cosh a + r \sinh a</math> where <math>r^2 = +1.</math>
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| Such elements arise in algebras of [[metric signature|mixed signature]], for example [[split-complex number]]s or [[split-quaternion]]s. It was the algebra of [[tessarines]] discovered by [[James Cockle (lawyer)|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.
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| The primary exponent of hyperbolic versors was [[Alexander Macfarlane]] as he worked to shape quaternion theory to serve physical science.<ref>''Papers on Space Analysis'' (1894), papers 2, 3, and 5, see external link below.</ref> He saw the modelling power of hyperbolic versors operating on the split-complex number plane, and in 1891 he introduced [[hyperbolic quaternion]]s to extend the concept to 4-space. Problems in that algebra led to use of [[biquaternion]]s after 1900. In a widely circulated review of 1899, Macfarlane said:
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| :…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.<ref>[[Science (journal)|Science]], 9:326 (1899)</ref>
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| 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.
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| In particular, for each ''r'' such that {{nowrap|1=''r r'' = +1}} or {{nowrap|1=''r r'' = −1}}, the mapping <math>a \mapsto \exp(ar)</math> takes the [[Real line#In real algebras|real line]] to a group of hyperbolic or ordinary versors. In the ordinary case, when ''r'' and −''r'' are [[antipodal point]]s 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)|doublet]].
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| 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]].
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| ==See also==
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| *[[Quaternions and spatial rotation]]
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| *[[Elliptic geometry #Elliptic space]]
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| *[[Rotations in 4-dimensional Euclidean space]]
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| ==References==
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| <references/>
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| * W.R. Hamilton (1899) ''Elements of Quaternions'', 2nd edition, edited by Charles Jasper Joly, Longmans Green & Company. See pp. 135–147.
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| * A.S. Hardy (1887) ''Elements of Quaternions'', pp. 71,2 "Representation of Versors by spherical arcs" and pp. 112–8 "Applications to Spherical Trigonometry".
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| * 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.
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| * Pieter Molenbroeck (1891) ''Theorie der Quaternionen'', Seite 48, "Darstellung der Versoren mittelst Bogen auf der Einheitskugel", Leiden: Brill.
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| ==External links==
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| * http://www.biology-online.org/dictionary/versor
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| * http://www.thefreedictionary.com/Versor
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| * A. Macfarlane (1894) ''[http://www.archive.org/details/principlesalgeb01macfgoog Papers on Space Analysis]'', B. Westerman, New York, weblink from [[archive.org]].
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| [[Category:Spherical trigonometry]]
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| [[Category:Quaternions]]
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| [[Category:Rotation in three dimensions]]
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