Cayley–Klein metric

The question recently arose in conversation whether a dissertation of 2 lines could deserve and get a Fellowship. ... Cayley's projective definition of length is a clear case if we may interpret "2 lines" with reasonable latitude. ... With Cayley the importance of the idea is obvious at first sight.

Littlewood (1986, pp. 39–40)

In mathematics, a Cayley–Klein metric is a metric on the complement of a fixed quadric in a projective space is defined using a cross-ratio. The construction originated with Arthur Cayley's essay "On the theory of distance" where he calls the quadric the absolute. The construction was developed in further detail by Felix Klein in papers in 1871 and 1873, and in his book Vorlesungen über Nicht-Euklidischen Geometrie(1928). The Cayley–Klein metrics are a unifying idea in geometry since the method is used to provide metrics in hyperbolic geometry, elliptic geometry, and Euclidean geometry. The field of non-Euclidean geometry rests largely on the footing provided by Cayley–Klein metrics.

Definition

Suppose that Q is a fixed quadric in projective space. If p and q are 2 points then the line through p and q intersects the quadric Q in two further points a and b. The Cayley–Klein distance d(p,q) from p to q is proportional to the logarithm of the cross-ratio:

$d(p,q)=C\log {\frac {|qa||bp|}{|pa||bq|}}$

for some fixed constant C.

Cayley–Klein geometry is the study of the group of motions that leave the Cayley–Klein metric invariant. This group is obtained as the collineations for which the absolute is stable. Indeed, cross-ratio is invariant under any collineation, and the stable absolute enables the metric comparison, which will be equality.

The extent of Cayley–Klein geometry was summarized by Horst and Rolf Struve in 2004:

There are three absolutes in the real projective line, seven in the real projective plane, and 18 in real projective space. All classical non-euclidean projective spaces as hyperbolic, elliptic, Galilean and Minkowskian and their duals can be defined this way.

Cayley-Klein Voronoi diagrams are affine diagrams with linear hyperplane bisectors.

Foundations

The algebra of throws is an approach to geometry that is independent of metric. Karl von Staudt's idea was to use the relation of projective harmonic conjugates as fundamental to the measure on a line. Klein made reference to Staudt's development, thus basing the new metric on logarithm, and the cross-ratio as a number generated by the geometric arrangement of four points. This procedure is necessary to avoid a circular definition of distance if cross-ratio is merely a double ratio of previously defined distances.

Cayley–Klein plane geometry depends upon the selection of a conic that becomes the absolute of the space. For example, the unit circle is the absolute of the Poincaré disk model and the Beltrami–Klein model in hyperbolic geometry. Similarly, the real line is the absolute of the Poincaré half-plane model.

For Euclidean space, Klein described the absolute as the "sphere-circle" consisting of the sphere of radius zero in the hyperplane at infinity:

$\lbrace (\xi ,\eta ,\zeta ,\tau ):\quad \tau =0,\quad \xi ^{2}+\eta ^{2}+\zeta ^{2}=0\rbrace \subset P^{3}(R).$

Klein also related the metric construction to physical science:

The case $x^{2}+y^{2}+z^{2}-t^{2}=0$ in the four-dimensional world or $dx^{2}+dy^{2}+dz^{2}-dt^{2}=0$ (to remain in three dimensions and use homogeneous coordinates) has recently won special significance through the relativity theory of physics.

Evidently Klein is referring here to $\left({\frac {dx}{dt}}\right)^{2}+\left({\frac {dy}{dt}}\right)^{2}+\left({\frac {dz}{dt}}\right)^{2}=1$ where physical velocities are bounded by the speed of light c, so that for any physical velocity v, the ratio v/c is confined to the interior of a unit sphere, and the surface of the sphere forms the Cayley absolute for the geometry

See also

• Hilbert metric

References

• Cayley, Arthur (1859), "A sixth memoir upon quantics", Philosophical Transactions of the Royal Society of London, 149: 61–90, doi:10.1098/rstl.1859.0004, ISSN 0080-4614, JSTOR 108690
• Klein, Felix (1871), "Über die sogenannte Nicht-Euklidische Geometrie", Mathematische Annalen (in German), 4, Springer Berlin / Heidelberg: 573–625, doi:10.1007/BF02100583, ISSN 0025-5831, JFM 03.0231.02
• Klein, Felix (1873), "Über die sogenannte Nicht-Euklidische Geometrie (Zweiter Aufsatz)", Mathematische Annalen, 6, Springer Berlin / Heidelberg: 112–145, doi:10.1007/BF01443189, ISSN 0025-5831, JFM 05.0271.01
• Harvey Lipkin (1985) Metrical Geometry from Georgia Institute of Technology
• Littlewood, J. E. (1986) [1953], Littlewood's miscellany, Cambridge University Press, ISBN 978-0-521-33058-9, MR 0872858
• Duncan Sommerville (1910/11) "Cayley–Klein metrics in n-dimensional space", Proceedings of the Edinburgh Mathematical Society 28:25–41.
• Nielsen, Frank; Muzellec, Boris; Nock, Richard (2016), "Classification with mixtures of curved Mahalanobis metrics", IEEE ICIP: 241–245, doi:10.1109/ICIP.2016.7532355
• Struve, Horst; Struve, Rolf (2004), "Projective spaces with Cayley–Klein metrics", Journal of Geometry, 81 (1): 155–167, doi:10.1007/s00022-004-1679-5, ISSN 0047-2468, MR 2134074
• Struve, Horst; Struve, Rolf (2010), "Non-euclidean geometries: the Cayley–Klein approach", Journal of Geometry, 89 (1): 151–170, doi:10.1007/s00022-010-0053-z, ISSN 0047-2468, MR 2739193
• Alfred North Whitehead (1898) Universal Algebra, Book VI Chapter 1: Theory of Distance, pp 347–70, especially Section 199 Cayley's Theory of Distance.