Cross-ratio

From Academic Kids

In mathematics, the cross-ratio cr( w, x, y, z ) of an ordered quadruple of complex numbers (which may be real numbers) is

<math>\operatorname{cr}(w,x,y,z)

=\frac{z-x}{z-w}\left/\frac{y-x}{y-w}\right..<math>

Cross-ratios are preserved by linear fractional transformations, i.e., functions of the form

<math>f(z)=\frac{az+b}{cz+d}.<math>

This fact is a special case of the fact that cross-ratios are preserved by projective transformations. For that reason, they play an important role in projective geometry. 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 is real if and only if the four complex numbers lie either on a common line or on a common circle.

Transformational approach

A more abstract definition uses the group action of the linear fractional transformations, to bring three of the four numbers into standard positions. There are three degrees of freedom in f, which contains constants a, b, c and d, but is unchanged when they are all multiplied by the same number. That means we can in practice send any three distinct numbers to any other three distinct given numbers. It is standard to take the given numbers as 0, 1, and the point at infinity (more concretely, make the denominator of f take the value 0).

Checking the formula

<math>\operatorname{cr}(w,x,y,z)

=\frac{z-x}{z-w}\left/\frac{y-x}{y-w}\right.,<math>

with x = 0 and y = 1 we get

<math>\operatorname{cr}(w,0,1,z)

= \frac{z(1-w)}{z-w}.<math>

If here we let w tend to infinity, the limit will be z. (It is quite possible to avoid the language of limits here, by using two homogeneous coordinates: this isn't really a question of analysis.)

We conclude that another description of the cross-ratio is this: assume given distinct w, x, y, z, and by a linear fractional transformation send w to the point at infinity, x to 0 and y to 1. Then the fourth number z is taken to a definite position λ. This is defined as the cross-ratio.

To be more accurate, as function of four unordered points, the cross-ratio could take one of six values, in general. Those correspond to the six permutations of the set {0, 1, point at infinity}. Since for example the linear fractional map g(z) = 1/z interchanges 0 and the point at infinity, while fixing 1, for a different ordering of the initial set of numbers we can get cross-ratio 1/λ. Similarly for h(z) = 1 − z; it interchanges 0 and 1, and takes cross-ratio λ to 1 − λ.

These two transformations generate the permutation group on the three points, since the two transpositions (12) and (13) generate the symmetric group on {1,2,3}. Sets of four points with certain special values of the cross-ratio are distinguished by having a non-trivial stabilizer in this permutation group. For example with λ = −1 the transformation g given above leaves the cross-ratio unchanged while w and x are switched over. This situation is what is called classically the harmonic cross-ratio.

Advanced points of view

From the point of view of group theory, this approach uses the fact that the group of linear fractional transformations PGL2 is triply transitive in acting on the projective line P1.

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

External links

it:Birapporto ru:Двойное отношение

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