Cayley–Klein metric

The metric distance between two points inside the absolute is the logarithm of the cross ratio formed by these two points and the two intersections of their line with the absolute

In mathematics, a Cayley–Klein metric is a metric on the complement of a fixed quadric in a projective space which is defined using a cross-ratio. The construction originated with Arthur Cayley's essay "On the theory of distance"[1] where he calls the quadric the absolute. The construction was developed in further detail by Felix Klein in papers in 1871 and 1873, and subsequent books and papers.[2] 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.

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]

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:

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

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

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

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

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

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:

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.

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.

Arthur Cayley (1859) defined the "absolute" upon which he based his projective metric as a general equation of a surface of second degree in terms of homogeneous coordinates:[1]

Felix Klein (1871) reformulated Cayley's expressions as follows: He wrote the absolute (which he called fundamental conic section) in terms of homogeneous coordinates:[31]

In his 1873 paper he pointed out the relation between the Cayley metric and transformation groups.[34] In particular, quadratic equations with real coefficients, corresponding to surfaces of second degree, can be transformed into a sum of squares, of which the difference between the number of positive and negative signs remains equal (this is now called Sylvester's law of inertia). If the sign of all squares is the same, the surface is imaginary with positive curvature. If one sign differs from the others, the surface becomes an ellipsoid or two-sheet hyperboloid with negative curvature.

In the first volume of his lectures on Non-Euclidean geometry in the winter semester 1889/90 (published 1892/1893), he discussed the Non-Euclidean plane, using these expressions for the absolute:[35]

and discussed their invariance with respect to collineations and Möbius transformations representing motions in Non-Euclidean spaces.

In the second volume containing the lectures of the summer semester 1890 (also published 1892/1893), Klein discussed Non-Euclidean space with the Cayley metric[36]

and went on to show that variants of this quaternary quadratic form can be brought into one of the following five forms by real linear transformations[37]