Cross-ratio

In geometry, the cross-ratio, also called the double ratio and anharmonic ratio, is a number associated with a list of four collinear points, particularly points on a projective line. Given four points A, B, C and D on a line, their cross ratio is defined as

where an orientation of the line determines the sign of each distance and the distance is measured as projected into Euclidean space. (If one of the four points is the line's point at infinity, then the two distances involving that point are dropped from the formula.) The point D is the harmonic conjugate of C with respect to A and B precisely if the cross-ratio of the quadruple is −1, called the harmonic ratio. The cross-ratio can therefore be regarded as measuring the quadruple's deviation from this ratio; hence the name anharmonic ratio.

The cross-ratio is preserved by linear fractional transformations. It is essentially the only projective invariant of a quadruple of collinear points; this underlies its importance for projective geometry.

The cross-ratio had been defined in deep antiquity, possibly already by Euclid, and was considered by Pappus, who noted its key invariance property. It was extensively studied in the 19th century.[1]

Variants of this concept exist for a quadruple of concurrent lines on the projective plane and a quadruple of points on the Riemann sphere. 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. In 1986 Alexander Jones made a translation of the original by Pappus, then wrote a commentary on how the lemmas of Pappus relate to modern terminology.[2]

Modern use of the cross ratio in projective geometry began with Lazare Carnot in 1803 with his book Géométrie de Position. The term used was le rapport anharmonique (Fr: anharmonic ratio). German geometers call it das Doppelverhältnis (Ger: double ratio).

Given three points on a line, a fourth point that makes the cross ratio equal to minus one is called the projective harmonic conjugate. In 1847 Carl von Staudt called the construction of the fourth point a throw (Wurf), and used the construction to exhibit arithmetic implicit in geometry. His Algebra of Throws provides an approach to numerical propositions, usually taken as axioms, but proven in projective geometry.[3]

The English term "cross-ratio" was introduced in 1878 by William Kingdon Clifford.[4]

The cross-ratio of a quadruple of distinct points on the real line with coordinates z1z2z3z4 is given by

It can also be written as a "double ratio" of two division ratios of triples of points:

In the notation of Euclidean geometry, if A, B, C, D are collinear points, their cross ratio is:

where each of the distances is signed according to a consistent orientation of the line.

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

The cross ratio of the four collinear points A, B, C, D can be written as

Four points can be ordered in 4! = 4 × 3 × 2 × 1 = 24 ways, but there are only six ways for partitioning them into two unordered pairs. Thus, four points can have only six different cross-ratios, which are related as:

The cross-ratio is a projective invariant in the sense that it is preserved by the projective transformations of a projective line.

In particular, if four points lie on a straight line L in R2 then their cross-ratio is a well-defined quantity, because any choice of the origin and even of the scale on the line will yield the same value of the cross-ratio.

If four collinear points are represented in homogeneous coordinates by vectors abcd such that c = a + b and d = ka + b, then their cross-ratio is k.[5]

Arthur Cayley and Felix Klein found an application of the cross-ratio to non-Euclidean geometry. Given a nonsingular conic C in the real projective plane, its stabilizer GC in the projective group G = PGL(3, R) acts transitively on the points in the interior of C. However, there is an invariant for the action of GC on pairs of points. In fact, every such invariant is expressible as a function of the appropriate cross ratio.[citation needed]

Explicitly, let the conic be the unit circle. For any two points P, Q, inside the unit circle . If the line connecting them intersects the circle in two points, X and Y and the points are, in order, X, P, Q, Y. Then the hyperbolic distance between P and Q in the Cayley–Klein model of the hyperbolic plane can be expressed as

(the factor one half is needed to make the curvature −1). 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. Being on a circle means the four points are the image of four real points under a Möbius transformation, and hence the cross ratio is a real number. The Poincaré half-plane model and Poincaré disk model are two models of hyperbolic geometry in the complex projective line.

These differ by the following permutations of the variables (in cycle notation):

For certain values of λ there will be greater symmetry and therefore fewer than six possible values for the cross-ratio. These values of λ correspond to fixed points of the action of S3 on the Riemann sphere (given by the above six functions); or, equivalently, those points with a non-trivial stabilizer in this permutation group.

The cross-ratio is invariant under the projective transformations of the line. In the case of a complex projective line, or the Riemann sphere, these transformations are known as Möbius transformations. A general Möbius transformation has the form

These transformations form a group acting on the Riemann sphere, the Möbius group.

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.

The action of the Möbius group is simply transitive on the set of triples of distinct points of the Riemann sphere: given any ordered triple of distinct points, (z2, z3, z4), there is a unique Möbius transformation f(z) that maps it to the triple (1, 0, ∞). This transformation can be conveniently described using the cross-ratio: since (z, z2, z3, z4) must equal (f(z), 1; 0, ∞), which in turn equals f(z), 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. The differences zjzk are invariant under the translations

where a is a constant in the ground field F. Furthermore, the division ratios are invariant under a homothety

for a non-zero constant b in F. Therefore, the cross-ratio is invariant under the affine transformations.

the affine line needs to be augmented by the point at infinity, denoted ∞, forming the projective line P1(F). Each affine mapping f : FF can be uniquely extended to a mapping of P1(F) into itself that fixes the point at infinity. The map T swaps 0 and ∞. The projective group is generated by T and the affine mappings extended to P1(F). In the case F = C, the complex plane, this results in the Möbius group. Since the cross-ratio is also invariant under T, it is invariant under any projective mapping of P1(F) into itself.

The concept of cross ratio only depends on the ring operations of addition, multiplication, and inversion (though inversion of a given element is not certain in a ring). One approach to cross ratio interprets it as a homography that takes three designated points to 0, 1, and infinity. Under restrictions having to do with inverses, it is possible to generate such a mapping with ring operations in the projective line over a ring. The cross ratio of four points is the evaluation of this homography at the fourth point.

The theory takes on a differential calculus aspect as the four points are brought into proximity. This leads to the theory of the Schwarzian derivative, and more generally of projective connections.

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.

Collinearity is not the only geometric property of configurations of points that must be maintained – for example, five points determine a conic, but six general points do not lie on a conic, so whether any 6-tuple of points lies on a conic is also a projective invariant. 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.