Cayley-Klein geometry

Cayley-Klein geometry is a method of constructing models of various non-Euclidean geometries. The two inputs are a projective space and an arbitrary quadric within that space which we call the absolute -- each pair of these determines a Cayley-Klein geometry. Given such a pair, the projective transformations which map points on the absolute back to points on the absolute are defined to be the congruences. Since we've defined a space and a set of congruences over it, we have defined a geometry. A measure of separation between points similar to a distance can then be derived, and a measure of separation between lines similar to an angle can also be derived. These measures of separation can be expressed using the projectively invariant notion of a cross ratio. The resulting measure of separation is not always a metric in the sense of metric spaces, and sometimes cannot be transformed into one; e.g. see for example Minkowski space.

Measures of separation

Consider a projective space over the field $\mathbb {R}$ and a quadric Q, which we call the absolute.

Given two points a and b, we can determine a measure of separation between them similar to a distance. We do so by joining a and b with a line. This line ought to meet the absolute at a pair of points p and q. To ensure that the line does indeed meet the absolute, the corresponding quadric is extended to complex projective space. Sometimes all the points on the quadric are non-real, and in that situation the points p and q will always be non-real. This also necessitates seeing the quadric as a symbolic expression, and not as a set. The measure of separation between a and b is then the cross-ratio (a,b;p,q).

Is the above measure of separation always a distance (in the sense of metric spaces)? It almost never is, but for some choices of quadric, there is a subset of projective space and a function $f:\mathbb {C} \to \mathbb {R} _{\geq 0}$ such that $f((a,b;p,q))$ is a metric. But for some quadrics, such a subset and function might not exist, and we cannot get a metric space.

Restricting the points and the lines

A projective space and an absolute are usually not understood as fully defining a Cayley-Klein geometry. In that sense, the introduction was misleading. An additional pair of inputs is needed: Some point $p$ in the ambient projective space and some line $\ell$ passing through $p$ . The points of a Cayley-Klein space are then defined to be the set of all points which $p$ can get mapped to by a congruence transformation. The lines of a Cayley-Klein space are similarly defined to be the set of lines which $\ell$ can get mapped to by a congruence transformation.

Consider the example where the absolute is a circle. There are in fact three Cayley-Klein geometries that this can define, depending on where we place $p$ and $\ell$ :

Hyperbolic plane, if we place $p$ inside the disk, and pick $\ell$ to be any line through $p$ .
De Sitter plane, if we place $p$ outside the disk, and pick $\ell$ to be any line through $p$ not meeting the circle.
Anti-de Sitter plane, if we place $p$ outside the disk, and pick $\ell$ to be any line through $p$ meeting the circle at two distinct points.

One could suggest other possibilities, like: Put the point $p$ outside the circle, and choose $\ell$ to go through $p$ and be tangent to the circle. Notice that in this example, the dimensionality of the space of points is 2 (as in our above three examples), but the dimensionality of the space of lines is only 1 (as the space of tangents to a circle forms only a 1-dimensional space). This example is pathological and motivates the following restriction: It's natural to insist that the dimensionality of the resulting space of points should equal that of the ambient projective space, and likewise the space of lines should have the same dimensionality as the projective space.

Examples

The following examples are not exhaustive.

Hyperbolic plane

We work over $\mathbb {RP} ^{2}$ . A quadric over this space is homogeneous in variables $x$ , $y$ and $z$ . The absolute for the Hyperbolic plane is defined by $x^{2}+y^{2}-z^{2}=0$ . We pick $p$ to be inside the circle, and pick $\ell$ to be any line through $p$ . This produces precisely the Beltrami–Klein model of the hyperbolic plane in which the absolute is clearly visible as a circle.

The hyperbolic distance between two points a and b is recovered from the cross ratio by ${\frac {1}{2}}\ln((a,b;p,q))$ . To ensure that we can indeed take this natural logarithm, we must ensure that the cross ratio is positive, which forces us to work in the space inside the circle. If we are not concerned about working over a metric space, we may do hyperbolic geometry on the absolute itself, and even over the region outside of it. This is not wrong because the set of congruences remains the same, and as a result so does the geometry.

Euclidean plane

We work over $\mathbb {RP} ^{2}$ . A quadric over this space is homogeneous in variables $x$ , $y$ and $z$ . The absolute for the Euclidean plane is defined by $x^{2}+y^{2}=0$ ; this is a degenerate quadric because the variable $z$ is missing. This absolute has no real points.

A line in $\mathbb {RP} ^{2}$ always meets this absolute at exactly two points, which are $[1:i:0]$ and $[1:-i:0]$ given using homogeneous coordinates. We call these points $I$ and $J$ respectively.

The Euclidean distance between two points a and b is recovered from the cross ratio by ${\frac {1}{2}}\ln((a,b;p,q))$ , which is equal to ${\frac {1}{2}}\ln((a,b;I,J))$ .

Elliptic plane

We work over $\mathbb {RP} ^{2}$ . A quadric over this space is homogeneous in variables $x$ , $y$ and $z$ . The absolute for the Elliptic plane is defined by $x^{2}+y^{2}+z^{2}=0$ . To complete the definition, $p$ can be chosen anywhere in $\mathbb {RP} ^{2}$ , and $\ell$ can be any line through $p$ . This absolute has no real points.

1+1 Minkowski space

We work over $\mathbb {RP} ^{2}$ . A quadric over this space is homogeneous in variables $x$ , $y$ and $z$ . The absolute for the Minkowski plane is defined by $x^{2}-y^{2}=0$ . This quadric is degenerate because the variable $z$ is missing.

This particular example is notable because a Minkowski space is never a metric space. The measure of separation, which can derived from the cross ratio, can never be transformed into a distance measure.

History

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.^[1] 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.^[2] 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.^[3]^[4] 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.^[5] This procedure is necessary to avoid a circular definition of distance if cross-ratio is merely a double ratio of previously defined distances.^[6] In particular, he showed that non-Euclidean geometries can be based on the Cayley–Klein metric.^[7]

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:^[8]

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.^[9]

References

^ Klein & Rosemann (1928), p. 163
^ Klein & Rosemann (1928), p. 138
^ Klein & Rosemann (1928), p. 303
^ Pierpont (1930), p. 67ff
^ Klein & Rosemann (1928), pp. 163, 304
^ Russell (1898), page 32
^ Campo & Papadopoulos (2014)
^ H & R Struve (2004) page 157
^ Nielsen (2016)

[1] Klein & Rosemann (1928), p. 163

[2] Klein & Rosemann (1928), p. 138

[3] Klein & Rosemann (1928), p. 303

[4] Pierpont (1930), p. 67ff

[5] Klein & Rosemann (1928), pp. 163, 304

[6] Russell (1898), page 32

[cam-7] Campo & Papadopoulos (2014)

[8] H & R Struve (2004) page 157

[9] Nielsen (2016)

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