Cayley–Klein metric

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 ⁡ | q a | | b p | | p a | | b q |

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:

{ ( ξ , η , ζ , τ ) : τ = 0 , ξ 2 + η 2 + ζ 2 = 0 } ⊂ 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 d x 2 + d y 2 + d z 2 − d t 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 ( d x d t ) 2 + ( d y d t ) 2 + ( d z d t ) 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