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. 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 quaternions provides a descriptive geometry of elliptic space in which Clifford parallelism is made explicit.

Introduction

The lines on 1 in elliptic space are described by versors with a fixed axis r:

$\lbrace e^{ar} :\ 0 \le a < \pi \rbrace$

For an arbitrary point u in elliptic space, two Clifford parallels to this line pass through u.

The right Clifford parallel is

$\lbrace u e^{ar}:\ 0 \le a < \pi \rbrace,$

and the left Clifford parallel is

$\lbrace e^{ar}u:\ 0 \le a < \pi \rbrace.$

Clifford surfaces

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.

Given two square roots of minus one in the quaternions, written r and s, the Clifford surface through them is given by

$\lbrace e^{ar}e^{br} :\ 0 \le a,b < \pi \rbrace.$

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 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, 189–93, Macmillan & Co..
• Guido Fubini (1900) D.H. Delphenich translator 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 12:57–78.
• J.A. Tyrrell & J.G. Semple (1971) Generalized Clifford Parallelism, Cambridge University Press ISBN 0-521-08042-8 .
• Duncan Sommerville (1914) The Elements of Non-Euclidean Geometry, page 108 Paratactic lines, George Bell & Sons