Cayley–Klein metric

The algebra of throws by Karl von Staudt (1847) is an approach to geometry that is independent of metric. The idea was to use the relation of projective harmonic conjugates and cross-ratios as fundamental to the measure on a line.[3] Another important insight was the Laguerre formula by Edmond Laguerre (1853), who showed that the Euclidean angle between two lines can be expressed as the logarithm of a cross-ratio.[4] Eventually, Cayley (1859) formulated relations to express distance in terms of a projective metric, and related them to general quadrics or conics serving as the absolute of the geometry.[5][6] Klein (1871, 1873) removed the last remnants of metric concepts from von Staudt's work and combined it with Cayley's theory, in order to base Cayley's new metric on logarithm and the cross-ratio as a number generated by the geometric arrangement of four points.[7] This procedure is necessary to avoid a circular definition of distance if cross-ratio is merely a double ratio of previously defined distances.[8] In particular, he showed that non-Euclidean geometries can be based on the Cayley–Klein metric.[9]

Cayley–Klein geometry is the study of the group of motions that leave the Cayley–Klein metric invariant. It depends upon the selection of a quadric or conic that becomes the absolute of the space. 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. 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.

The extent of Cayley–Klein geometry was summarized by Horst and Rolf Struve in 2004:[10]

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.[11]

Suppose that Q is a fixed quadric in projective space that becomes the absolute of that geometry. If a and b are 2 points then the line through a and b intersects the quadric Q in two further points p and q. The Cayley–Klein distance d(a,b) from a to b is proportional to the logarithm of the cross-ratio:[12]

${\displaystyle d(a,b)=C\log {\frac {|bp||qa|}{|ap||qb|}}}$ for some fixed constant C.

When C is real, it represents the hyperbolic distance of hyperbolic geometry, when imaginary it relates to elliptic geometry. The absolute can also be expressed in terms of arbitrary quadrics or conics having the form in homogeneous coordinates:

${\displaystyle \Omega =\sum \omega _{\alpha \beta }x_{\alpha }x_{\beta }=0,\ \left(\omega _{\alpha \beta }=\omega _{\beta \alpha }\right)}$

(where α,β=1,2,3 relates to the plane and α,β=1,2,3,4 to space), thus:[13]

${\displaystyle {\begin{matrix}{\begin{aligned}d&=C\log {\frac {\Omega _{xy}+{\sqrt {\Omega _{xy}^{2}-\Omega _{xx}\Omega _{yy}}}}{\Omega _{xy}-{\sqrt {\Omega _{xy}^{2}-\Omega _{xx}\Omega _{yy}}}}}\\&=2iC\cdot \operatorname {acos} {\frac {\Omega _{xy}}{\sqrt {\Omega _{xx}\cdot \Omega _{yy}}}}\end{aligned}}\\\hline {\begin{matrix}\Omega _{xx}=\sum \omega _{\alpha \beta }x_{\alpha }x_{\beta }=0\\\Omega _{yy}=\sum \omega _{\alpha \beta }y_{\alpha }y_{\beta }=0\\\Omega _{xy}=\sum \omega _{\alpha \beta }x_{\alpha }y_{\beta }\end{matrix}}\end{matrix}}}$

The corresponding hyperbolic distance is (with C=1/2 for simplification):[14]

${\displaystyle d=\operatorname {acosh} {\frac {\Omega _{xy}}{\sqrt {\Omega _{xx}\cdot \Omega _{yy}}}}}$

or in elliptic geometry (with C=i/2 for simplification)[15]

${\displaystyle d=\operatorname {acos} {\frac {\Omega _{xy}}{\sqrt {\Omega _{xx}\cdot \Omega _{yy}}}}}$

Any quadric (or surface of second order) with real coefficients of the form ${\displaystyle \Omega =\sum \omega _{\alpha \beta }x_{\alpha }x_{\beta }=0}$ can be transformed into normal or canonical forms in terms of sums of squares, while the difference in the number of positive and negative signs doesn't change under a real homogeneous transformation of determinant ≠0 by Sylvester's law of inertia, with the following classification ("zero-part" means real equation of the quadric, but no real points):[16]

I. Proper surfaces of second order.

1. ${\displaystyle x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}=0}$. Zero-part surface.

2. ${\displaystyle x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-x_{4}^{2}=0}$. Oval surface.

a) Ellipsoid

b) Elliptic paraboloid

c) Two-sheet hyperboloid

3. ${\displaystyle x_{1}^{2}+x_{2}^{2}-x_{3}^{2}-x_{4}^{2}=0}$. Ring surface.

a) One-sheet hyperboloid

b) Hyperbolic paraboloid

II. Conic surfaces of second order.

1. ${\displaystyle x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=0}$. Zero-part cone.

a) Zero-part cone

b) Zero-part cylinder

2. ${\displaystyle x_{1}^{2}+x_{2}^{2}-x_{3}^{2}=0}$. Ordinary cone.

a) Cone

b) Elliptic cylinder

c) Parabolic cylinder

d) Hyperbolic cylinder

III. Plane pairs.

1. ${\displaystyle x_{1}^{2}+x_{2}^{2}=0}$. Conjugate imaginary plane pairs.

a) Mutually intersecting imaginary planes.

b) Parallel imaginary planes.

2. ${\displaystyle x_{1}^{2}-x_{2}^{2}=0}$. Real plane pairs.

a) Mutually intersecting planes.

b) Parallel planes.

c) One plane is finite, the other one infinitely distant, thus not existent from the affine point of view.

IV. Double counting planes.

1. ${\displaystyle x_{1}^{2}=0}$.

a) Double counting finite plane.

b) Double counting infinitely distant plane, not existent in affine geometry.

The collineations leaving invariant these forms can be related to linear fractional transformations or Möbius transformations.[17] Such forms and their transformations can now be applied to several kinds of spaces, which can be unified by using a parameter ε (where ε=0 for Euclidean geometry, ε=1 for elliptic geometry, ε=-1 for hyperbolic geometry), so that the equation in the plane becomes ${\displaystyle x_{1}^{2}+x_{2}^{2}+{\tfrac {1}{\varepsilon }}x_{3}^{2}=0}$[18] and in space ${\displaystyle x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+{\tfrac {1}{\varepsilon }}x_{4}^{2}=0}$.[19] For instance, the absolute for the Euclidean plane can now be represented by ${\displaystyle x_{1}^{2}+x_{2}^{2}=0,\ x_{3}=0}$.[20]

The elliptic plane or space is related to zero-part surfaces in homogeneous coordinates:[21]

${\displaystyle {\begin{array}{c|c}\Omega =x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=0&\Omega =x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}=0\\\hline d=\operatorname {acos} {\frac {x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3}}{{\sqrt {x_{1}^{2}+x_{2}^{2}+x_{3}^{2}}}{\sqrt {y_{1}^{2}+y_{2}^{2}+y_{3}^{2}}}}}&d=\operatorname {acos} {\frac {x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3}+x_{4}y_{4}}{{\sqrt {x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}}}{\sqrt {y_{1}^{2}+y_{2}^{2}+y_{3}^{2}+x_{4}^{2}}}}}\end{array}}}$

or using inhomogeneous coordinates ${\displaystyle \left[{\mathfrak {x}},{\mathfrak {y}},\dots ,1\right]=\left[{\tfrac {x_{1}}{x_{n}}},{\tfrac {x_{2}}{x_{n}}},\dots ,{\tfrac {x_{n}}{x_{n}}}\right]}$ by which the absolute becomes the imaginary unit circle or unit sphere:[22]

${\displaystyle {\begin{array}{c|c}\Omega ={\mathfrak {x}}^{2}+{\mathfrak {y}}^{2}+1=0&\Omega ={\mathfrak {x}}^{2}+{\mathfrak {y}}^{2}+{\mathfrak {z}}^{2}+1=0\\\hline d=\operatorname {acos} {\frac {{\mathfrak {x}}_{1}{\mathfrak {x}}_{2}+{\mathfrak {y}}_{1}{\mathfrak {y}}_{2}+1}{{\sqrt {{\mathfrak {x}}_{1}^{2}+{\mathfrak {y}}_{1}^{2}+1}}{\sqrt {{\mathfrak {x}}_{2}^{2}+{\mathfrak {y}}_{2}^{2}+1}}}}&d=\operatorname {acos} {\frac {{\mathfrak {x}}_{1}{\mathfrak {x}}_{2}+{\mathfrak {y}}_{1}{\mathfrak {y}}_{2}+{\mathfrak {z}}_{1}{\mathfrak {z}}_{2}+1}{{\sqrt {{\mathfrak {x}}_{1}^{2}+{\mathfrak {y}}_{1}^{2}+{\mathfrak {z}}_{1}^{2}+1}}{\sqrt {{\mathfrak {x}}_{2}^{2}+{\mathfrak {y}}_{2}^{2}+{\mathfrak {y}}_{1}^{2}+1}}}}\end{array}}}$

or expressing the homogeneous coordinates in terms of the condition ${\displaystyle x_{1}^{2}+\dots +x_{n}^{2}=y_{1}^{2}+\dots +y_{n}^{2}=1}$ (Weierstrass coordinates) the distance simplifies to:[23]

${\displaystyle {\begin{array}{c|c}d=\operatorname {acos} \left(x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3}\right)&d=\operatorname {acos} \left(x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3}+x_{4}y_{4}\right)\end{array}}}$

The hyperbolic plane or space is related to the oval surface in homogeneous coordinates:[24]

${\displaystyle {\begin{array}{c|c}\Omega =x_{1}^{2}+x_{2}^{2}-x_{3}^{2}=0&\Omega =x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-x_{4}^{2}=0\\\hline d=\operatorname {acosh} {\frac {x_{1}y_{1}+x_{2}y_{2}-x_{3}y_{3}}{{\sqrt {x_{1}^{2}+x_{2}^{2}-x_{3}^{2}}}{\sqrt {y_{1}^{2}+y_{2}^{2}-y_{3}^{2}}}}}&d=\operatorname {acosh} {\frac {x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3}-x_{4}y_{4}}{{\sqrt {x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-x_{4}^{2}}}{\sqrt {y_{1}^{2}+y_{2}^{2}+y_{3}^{2}-x_{4}^{2}}}}}\end{array}}}$

or using inhomogeneous coordinates ${\displaystyle \left[{\mathfrak {x}},{\mathfrak {y}},\dots ,1\right]=\left[{\tfrac {x_{1}}{x_{n}}},{\tfrac {x_{2}}{x_{n}}},\dots ,{\tfrac {x_{n}}{x_{n}}}\right]}$ by which the absolute becomes the unit circle or unit sphere:[25]

${\displaystyle {\begin{array}{c|c}\Omega ={\mathfrak {x}}^{2}+{\mathfrak {y}}^{2}-1=0&\Omega ={\mathfrak {x}}^{2}+{\mathfrak {y}}^{2}+{\mathfrak {z}}^{2}-1=0\\\hline d=\operatorname {acosh} {\frac {{\mathfrak {x}}_{1}{\mathfrak {x}}_{2}+{\mathfrak {y}}_{1}{\mathfrak {y}}_{2}-1}{{\sqrt {{\mathfrak {x}}_{1}^{2}+{\mathfrak {y}}_{1}^{2}-1}}{\sqrt {{\mathfrak {x}}_{2}^{2}+{\mathfrak {y}}_{2}^{2}-1}}}}&d=\operatorname {acosh} {\frac {{\mathfrak {x}}_{1}{\mathfrak {x}}_{2}+{\mathfrak {y}}_{1}{\mathfrak {y}}_{2}+{\mathfrak {z}}_{1}{\mathfrak {z}}_{2}-1}{{\sqrt {{\mathfrak {x}}_{1}^{2}+{\mathfrak {y}}_{1}^{2}+{\mathfrak {z}}_{1}^{2}-1}}{\sqrt {{\mathfrak {x}}_{2}^{2}+{\mathfrak {y}}_{2}^{2}+{\mathfrak {y}}_{1}^{2}-1}}}}\end{array}}}$

or expressing the homogeneous coordinates in terms of the condition ${\displaystyle x_{1}^{2}+x_{2}^{2}+\dots -x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots -y_{n}^{2}=-1}$ (Weierstrass coordinates of the hyperboloid model) the distance simplifies to:[26]

${\displaystyle {\begin{array}{c|c}d=\operatorname {acosh} \left(x_{1}y_{1}+x_{2}y_{2}-x_{3}y_{3}\right)&d=\operatorname {acosh} \left(x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3}-x_{4}y_{4}\right)\end{array}}}$

In his lectures on the history of mathematics from 1919/20, published posthumously 1926, Klein wrote:[27]

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

That is, the absolutes ${\displaystyle x_{1}^{2}+x_{2}^{2}-x_{3}^{2}=0}$ or ${\displaystyle x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-x_{4}^{2}=0}$ in hyperbolic geometry (as discussed above), correspond to the intervals ${\displaystyle x^{2}+y^{2}-t^{2}=0}$ or ${\displaystyle x^{2}+y^{2}+z^{2}-t^{2}=0}$ in spacetime, and its transformation leaving the absolute invariant can be related to Lorentz transformations. Similarly, the equations of the unit circle or unit sphere in hyperbolic geometry correspond to physical velocities ${\displaystyle \left({\tfrac {dx}{dt}}\right)^{2}+\left({\tfrac {dy}{dt}}\right)^{2}=1}$ or ${\displaystyle \left({\tfrac {dx}{dt}}\right)^{2}+\left({\tfrac {dy}{dt}}\right)^{2}+\left({\tfrac {dz}{dt}}\right)^{2}=1}$ in relativity, which 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.

Additional details about the relation between the Cayley–Klein metric for hyperbolic space and Minkowski space of special relativity were pointed out by Klein in 1910,[28] as well as in the 1928 edition of his lectures on non-Euclidean geometry.[29]

In 2008 Horst Martini and Margarita Spirova generalized the first of Clifford's circle theorems and other Euclidean geometry using affine geometry associated with the Cayley absolute:

If the absolute contains a line, then one obtains a subfamily of affine Cayley-Klein geometries. If the absolute consists of a line f and a point F on f, then we have the isotropic geometry. An isotropic circle is a conic touching f at F.[30]

Use homogeneous coordinates (x,y,z). Line f at infinity is z = 0. If F = (0,1,0), then a parabola with diameter parallel to y-axis is an isotropic circle.

Let P = (1,0,0) and Q = (0,1,0) be on the absolute, so f is as above. A rectangular hyperbola in the (x,y) plane is considered to pass through P and Q on the line at infinity. These curves are the pseudo-Euclidean circles.

The treatment by Martini and Spirova uses dual numbers for the isotropic geometry and split-complex numbers for the pseudo-Euclidean geometry. These generalized complex numbers associate with their geometries as ordinary complex numbers do with Euclidean geometry.

• Hilbert metric

Historical

• von Staudt, K. (1847). Geometrie der Lage. Nürnberg: Nürnberg F. Korn.
• Laguerre, E. (1853). "Note sur la théorie des foyers". Nouvelles annales de mathématiques. 12: 57–66.
• Cayley, A. (1859). "A sixth memoir upon quantics". Philosophical Transactions of the Royal Society of London. 149: 61–90. doi:10.1098/rstl.1859.0004.
• Klein, F. (1871). "Ueber die sogenannte Nicht-Euklidische Geometrie". Mathematische Annalen. 4 (4): 573–625. doi:10.1007/BF02100583.
• Klein, F. (1873). "Ueber die sogenannte Nicht-Euklidische Geometrie". Mathematische Annalen. 6 (2): 112–145. doi:10.1007/BF01443189.
• Klein, F. (1893a). Schilling, Fr. (ed.). Nicht-Euklidische Geometrie I, Vorlesung gehalten während des Wintersemesters 1889–90. Göttingen. (second print, first print in 1892)
• Klein, F. (1893b). Schilling, Fr. (ed.). Nicht-Euklidische Geometrie II, Vorlesung gehalten während des Sommersemesters 1890. Göttingen. (second print, first print in 1892)

Secondary sources

• Killing, W. (1885). Die nicht-euklidischen Raumformen. Leipzig: Teubner.
• Fricke, R.; Klein, F. (1897). Vorlesungen über die Theorie der automorphen Functionen – Erster Band: Die gruppentheoretischen Grundlagen. Leipzig: Teubner.
• Bertrand Russell (1898) An Essay on the Foundations of Geometry, re-issued 1956 by Dover Books
• Alfred North Whitehead (1898) Universal Algebra, Book VI Chapter 1: Theory of Distance, pp 347–70, especially Section 199 Cayley's Theory of Distance.
• Hausdorff, F. (1899). "Analytische Beiträge zur nichteuklidischen Geometrie". Leipziger Math.-Phys. Berichte. 51: 161–214.
• Duncan Sommerville (1910/11) "Cayley–Klein metrics in n-dimensional space", Proceedings of the Edinburgh Mathematical Society 28:25–41.
• Klein, Felix (1910). "Über die geometrischen Grundlagen der Lorentzgruppe" . Jahresbericht der Deutschen Mathematiker-Vereinigung. 19: 533–552. doi:10.1007/978-3-642-51960-4_31. ISBN 978-3-642-51898-0. Reprinted in Klein, Felix (1921). Gesammelte mathematische Abhandlungen. 1. pp. 533–552. doi:10.1007/978-3-642-51960-4_31. English translation by David Delphenich: On the geometric foundations of the Lorentz group
• Veblen, O. and Young J.W. (1918). Projective geometry. Boston: Ginn.
• Liebmann, H. (1923). Nichteuklidische Geometrie. Berlin & Leipzig: Berlin W. de Gruyter.
• Klein, F. (1926). Courant, R.; Neugebauer, O. (eds.). Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert. Berlin: Springer.; English translation: Development of Mathematics in the 19th Century by M. Ackerman, Math Sci Press
• Klein, F. (1928). Rosemann, W. (ed.). Vorlesungen über nicht-Euklidische Geometrie. Berlin: Springer.
• Pierpont, J. (1930). "Non-euclidean geometry, a retrospect". Bulletin of the American Mathematical Society. 36 (2): 66–76. doi:10.1090/S0002-9904-1930-04885-5.
• Littlewood, J. E. (1986) [1953], Littlewood's miscellany, Cambridge University Press, ISBN 978-0-521-33058-9, MR 0872858
• Harvey Lipkin (1985) Metrical Geometry from Georgia Institute of Technology
• 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
• Martini Horst, Spirova Margarita (2008). "Circle geometry in affine Cayley-Klein planes". Periodica Mathematica Hungarica. 57 (2): 197–206. doi:10.1007/s10998-008-8197-5.
• 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
• A’Campo, N.; Papadopoulos, A. (2014). "On Klein's So-called Non-Euclidean geometry". In Ji, L.; Papadopoulos, A. (eds.). Sophus Lie and Felix Klein: The Erlangen Program and Its Impact in Mathematics and Physics. pp. 91–136. arXiv:1406.7309. doi:10.4171/148-1/5. ISBN 978-3-03719-148-4.
• Nielsen, Frank; Muzellec, Boris; Nock, Richard (2016), "Classification with mixtures of curved mahalanobis metrics", 2016 IEEE International Conference on Image Processing (ICIP), pp. 241–245, doi:10.1109/ICIP.2016.7532355, ISBN 978-1-4673-9961-6

• Jan Drösler (1979) "Foundations of multidimensional metric scaling in Cayley-Klein geometries", British Journal of Mathematical and Statistical Psychology 32(2); 185–211