Cross-ratio

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In mathematics, the cross-ratio of a set of four distinct points on the complex plane is given by

$(z_1,z_2;z_3,z_4) = \frac{(z_1-z_3)(z_2-z_4)}{(z_1-z_4)(z_2-z_3)}.$

This definition can be extended to the entire Riemann sphere (i.e. the complex plane plus the point at infinity) by continuity.

More generally, the cross-ratio can be defined on any projective line (the Riemann sphere is just the complex projective line). It is given by the above expression in any affine coordinate chart. Cross-ratios are invariants of projective geometry in the sense that they are preserved by projective transformations.

Historically it was noticed that if four lines in the plane pass through a point P, and a fifth line L not through P crosses them in four points, then the cross-ratio of the directed lengths on L formed by the four points taken in order was independent of L. That is, it is an invariant of the system of four lines.

The cross-ratio of four complex numbers is real if and only if the four numbers are either collinear or concyclic.

1 Symmetry
2 Transformational approach
3 Advanced points of view
4 External links

[edit] Symmetry

There are different definitions of the cross-ratio used in the literature. However, they all differ from each other by a some possible permutation of the coordinates. In general, there are six possible different values the cross-ratio can take depending on the order in which the points z_i are given. Since there are 24 possible permutations of the four coordinates, some permutations must leave the cross-ratio unaltered. In fact, exchanging any two pairs of coordinates preserves the cross-ratio:

$(z_1,z_2;z_3,z_4) = (z_2,z_1;z_4,z_3) = (z_3,z_4;z_1,z_2) = (z_4,z_3;z_2,z_1).\,$

Using these symmetries, there can then be 6 possible values of the cross-ratio, depending on the order in which the points are given. These are:

$(z_1, z_2; z_3, z_4) = \lambda\,$		$(z_1, z_2; z_4, z_3) = {1\over\lambda}$
$(z_1, z_3; z_4, z_2) = {1\over{1-\lambda}}$		$(z_1, z_3; z_2, z_4) = 1-\lambda\,$
$(z_1, z_4; z_3, z_2) = {\lambda\over{\lambda-1}}$		$(z_1, z_4; z_2, z_3) = {{\lambda-1}\over\lambda}$

In the language of group theory, the symmetric group S₄ acts on the cross-ratio by permuting coordinates. The kernel of this action is the Klein four-group (this in the group which preserves the cross-ratio). The effective symmetry group is then the quotient group which is isomorphic to S₃.

For certain values of λ there will be an enhanced symmetry and therefore less 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 first set of fixed points is {0, 1, ∞}. However, the cross-ratio can never take on these values if the points {z_i} are all distinct. These values are limit values as any pair of coordinates approach each other:

$(z,z_2;z,z_4) = (z_1,z;z_3,z) = 0\,$

$(z,z;z_3,z_4) = (z_1,z_2;z,z) = 1\,$

$(z,z_2;z_3,z) = (z_1,z;z,z_4) = \infty\,$

The second set of fixed points is {−1, 1/2, 2}. This situation is what is classically called the harmonic cross-ratio. The most symmetric cross-ratio occurs when $\lambda = e^{\pm i\pi/3}$ . These are then the only two possible values of the cross-ratio.

[edit] Transformational approach

Cross-ratios are preserved by projective transformations of the Riemann sphere, also known as Möbius transformations. A generic Möbius transformation is given by

$f(z) = \frac{az+b}{cz+d}\;,\quad ad-bc \ne 0$

To say that this preserves the cross-ratio means that

$(f(z_1), f(z_2); f(z_3), f(z_4)) = (z_1, z_2; z_3, z_4).\,$

Acting on the Riemann sphere, the group of Möbius transformations has the property that there is a unique Möbius transformation taking any set of three points onto any other set of three points (i.e. the action is sharply 3-transitive). Therefore, given four points on the Riemann sphere, we can find a unique transformation which takes three of those points to the points 0, 1, and ∞. The destination of the fourth points turns out to be related to the cross-ratio of the original points.

To see this, note that

$(z,1;0,\infty) = \lim_{w\to\infty} \frac{z(1-w)}{z-w} = z.\,$

Therefore given four points $(z 1, z 2; z 3, z 4)$ we can find a unique transformation f which sends

$z_2 \to 1,\; z_3 \to 0,\; z_4 \to \infty$

The point $z 1$ will then get sent to the cross-ratio $(z 1, z 2; z 3, z 4) = f (z 1).$ Looked at in a different light, the cross-ratio, thought of as a function of $z 1$ , is the unique Möbius transformation taking the points $(z 2, z 3, z 4)$ to $(1,0,\infty)$ .

[edit] Advanced points of view

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. These ideas are applied to conformal field theory.