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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 geometry#Elliptic space|elliptic space]]. Since [[parallel lines]] have the property of equidistance, the term "parallel" was appropriated from [[Euclidean geometry]], but in fact the "lines" of elliptic geometry are curves, and they have finite length, unlike the lines of Euclidean geometry. The algebra of [[quaternion]]s provides a descriptive geometry of elliptic space in which Clifford parallelism is made explicit.
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==Introduction==
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The lines on 1 in elliptic space are described by [[versor]]s with a fixed axis ''r'':
:<math>\lbrace e^{ar} :\ 0 \le a < \pi \rbrace</math>


For an arbitrary point ''u'' in elliptic space, two Clifford parallels to this line pass through ''u''.
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The right Clifford parallel is
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:<math>\lbrace u e^{ar}:\ 0 \le a < \pi \rbrace,</math>


and the left Clifford parallel is
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:<math>\lbrace e^{ar}u:\ 0 \le a < \pi \rbrace.</math>


==Clifford surfaces==
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Rotating a line about another, to which is Clifford parallel, creates a Clifford surface.


The Clifford parallels through points on the surface all lie in the surface. A Clifford surface is thus a [[ruled surface]] since every point is on two lines, each contained in the surface.
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Given two square roots of minus one in the [[quaternion]]s, written ''r'' and ''s'', the Clifford surface through them is given by
:<math>\lbrace e^{ar}e^{br} :\ 0 \le a,b < \pi \rbrace.</math>
 
==History==
Clifford parallels were first described in 1873 by the English mathematician [[William Kingdon Clifford]].
 
In 1900 [[Guido Fubini]] wrote his doctoral thesis on ''Clifford's parallelism in elliptic spaces''. Two years later [[Luigi Bianchi|Bianchi]] discussed Fubini's thesis in a widely read work on [[differential geometry]].
 
In 1931 [[Heinz Hopf]] used Clifford parallels to construct the [[Hopf map]].
 
==See also==
*[[Clifford torus]]
 
==References==
* [[William Kingdon Clifford]] (1882) ''Mathematical Papers'', 189&ndash;93, [[Macmillan & Co.]].
* [[Guido Fubini]] (1900) D.H. Delphenich translator [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/fubini_-_clifford_parallelism.pdf Clifford Parallelism in Elliptic Spaces], Laurea thesis, Pisa.
* Laptev, B.L. & B.A. Rozenfel'd (1996) ''Mathematics of the 19th Century: Geometry'', page 74, [[Birkhäuser Verlag]] ISBN 3-7643-5048-2 .
*[[Georges Lemaître]] (1948) "Quaternions et espace elliptique", ''Acta'' [[Pontifical Academy of Sciences]] 12:57&ndash;78.
* J.A. Tyrrell & J.G. Semple (1971) ''Generalized Clifford Parallelism'', [[Cambridge University Press]] ISBN 0-521-08042-8 .
 
 
[[Category:Non-Euclidean geometry]]

Latest revision as of 12:55, 10 December 2014

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