Cross-ratio

The cross-ratio had been defined in deep antiquity, possibly already by Euclid, and was considered by Pappus, who noted its key invariance property.

In the Cayley–Klein model of hyperbolic geometry, the distance between points is expressed in terms of a certain cross-ratio.

Pappus of Alexandria made implicit use of concepts equivalent to the cross-ratio in his Collection: Book VII.

Early users of Pappus included Isaac Newton, Michel Chasles, and Robert Simson.

[2] Modern use of the cross ratio in projective geometry began with Lazare Carnot in 1803 with his book Géométrie de Position.

[3][pages needed] Chasles coined the French term rapport anharmonique [anharmonic ratio] in 1837.

Carl von Staudt was unsatisfied with past definitions of the cross-ratio relying on algebraic manipulation of Euclidean distances rather than being based purely on synthetic projective geometry concepts.

In 1847, von Staudt demonstrated that the algebraic structure is implicit in projective geometry, by creating an algebra based on construction of the projective harmonic conjugate, which he called a throw (German: Wurf): given three points on a line, the harmonic conjugate is a fourth point that makes the cross ratio equal to −1.

His algebra of throws provides an approach to numerical propositions, usually taken as axioms, but proven in projective geometry.

If the displacements themselves are taken to be signed real numbers, then the cross ratio between points can be written If

The same formulas can be applied to four distinct complex numbers or, more generally, to elements of any field, and can also be projectively extended as above to the case when one of them is

As long as the points A, B, C, and D are distinct, the cross ratio (A, B; C, D) will be a non-zero real number.

It turns out that the cross-ratio of these points (taken in a fixed order) does not depend on the choice of a line

[7] Arthur Cayley and Felix Klein found an application of the cross-ratio to non-Euclidean geometry.

Since the cross-ratio is invariant under projective transformations, it follows that the hyperbolic distance is invariant under the projective transformations that preserve the conic C. Conversely, the group G acts transitively on the set of pairs of points (p, q) in the unit disk at a fixed hyperbolic distance.

Later, partly through the influence of Henri Poincaré, the cross ratio of four complex numbers on a circle was used for hyperbolic metrics.

Thus, the other permutations of the four variables alter the cross-ratio to give the following six values, which are the orbit of the six-element group

It is convenient to visualize this by a Möbius transformation M mapping the real axis to the complex unit circle (the equator of the Riemann sphere), with 0, 1, ∞ equally spaced.Considering {0, 1, ∞} as the vertices of the dihedron, the other fixed points of the 2-cycles are the points {2, −1, 1/2}, which under M are opposite each vertex on the Riemann sphere, at the midpoint of the opposite edge.

these are examples of Möbius transformations, which under composition of functions form the Mobius group PGL(2, C).

Algebraically, this corresponds to the action of S3 on the 2-cycles (its Sylow 2-subgroups) by conjugation and realizes the isomorphism with the group of inner automorphisms,

It may also be realised as the six Möbius transformations mentioned,[8] which yields a projective representation of S3 over any field (since it is defined with integer entries), and is always faithful/injective (since no two terms differ only by 1/−1).

The projective invariance of the cross-ratio means that The cross-ratio is real if and only if the four points are either collinear or concyclic, reflecting the fact that every Möbius transformation maps generalized circles to generalized circles.

, we obtain An alternative explanation for the invariance of the cross-ratio is based on the fact that the group of projective transformations of a line is generated by the translations, the homotheties, and the multiplicative inversion.

In order to obtain a well-defined inversion mapping the affine line needs to be augmented by the point at infinity, denoted

, then the real part of the cross ratio is given by: This is an invariant of the 2-dimensional special conformal transformation such as inversion

One approach to cross ratio interprets it as a homography that takes three designated points to 0, 1, and ∞.

The theory takes on a differential calculus aspect as the four points are brought into proximity.

The cross-ratio does not generalize in a simple manner to higher dimensions, due to other geometric properties of configurations of points, notably collinearity – configuration spaces are more complicated, and distinct k-tuples of points are not in general position.

One can study orbits of points in general position – in the line "general position" is equivalent to being distinct, while in higher dimensions it requires geometric considerations, as discussed – but, as the above indicates, this is more complicated and less informative.

However, a generalization to Riemann surfaces of positive genus exists, using the Abel–Jacobi map and theta functions.

Points A , B , C , D and A ′ , B ′ , C ′ , D ′ are related by a projective transformation so their cross ratios, ( A , B ; C , D ) and ( A ′ , B ′ ; C ′ , D ′ ) are equal.
D is the harmonic conjugate of C with respect to A and B , so that the cross-ratio ( A , B ; C , D ) equals −1 .
Use of cross-ratios in projective geometry to measure real-world dimensions of features depicted in a perspective projection . A, B, C, D and V are points on the image, their separation given in pixels; A', B', C' and D' are in the real world, their separation in metres.
1. The width of the side street, W is computed from the known widths of the adjacent shops.
2. As a vanishing point , V is visible, the width of only one shop is needed.

The stabilizer of {0, 1, ∞} is isomorphic to the rotation group of the trigonal dihedron , the dihedral group D 3 . It is convenient to visualize this by a Möbius transformation M mapping the real axis to the complex unit circle (the equator of the Riemann sphere ), with 0, 1, ∞ equally spaced.

Considering {0, 1, ∞} as the vertices of the dihedron, the other fixed points of the 2 -cycles are the points {2, −1, 1/2}, which under M are opposite each vertex on the Riemann sphere, at the midpoint of the opposite edge. Each 2 -cycles is a half-turn rotation of the Riemann sphere exchanging the hemispheres (the interior and exterior of the circle in the diagram).

The fixed points of the 3 -cycles are exp(± iπ /3) , corresponding under M to the poles of the sphere: exp( iπ /3) is the origin and exp(− iπ /3) is the point at infinity . Each 3 -cycle is a 1/3 turn rotation about their axis, and they are exchanged by the 2 -cycles.