Cross-ratio

The cross ratio is in the geometry of a number indicating the relative position of four different points lying on a straight line. Two of the points ( for example, A and B), thereby determine a distance ( such as T and U ) is divided by the other two points. The double ratio ( Abtu ) is now defined as the ratio of the two ratios ( ABT) and ( ABU ):

This definition implies that the cross ratio is positive if the points of T and U are either both outside the segment [ AB ] within the segment [ AB ], or both. If one of the points of division within and the other outside the segment [ AB ], so the cross-ratio ( Abtu ) is negative.

Analytic Geometry

In a parallel coordinate system are given by the four points A ( a | a ') and B ( B | B') and T (t | t ' ) and U ( u | u' ), then

As with the division ratio thus results the double ratio of the cross-ratio of the corresponding coordinate sections.

Swapping the generating points

Reverses to the underlying track with the track formed by the points of division, then the cross-ratio does not change:

Exchanges If, however, only the endpoints of the output line, the result is the reciprocal value of the double ratio:

Reverses to the endpoint of the base point with the first part, we obtain

Four different points on a line so six form (not necessarily all different ) cross-ratios:

Harmonic division

From a harmonic pitch is when the points T and U share the segment [ AB ] inside and outside at the same rate, so if

Is.

The cross ratio is then

If you want to three given points on a line g the fourth determine, so that these parts harmoniously, as one proceeds as follows ( the points A, C, B ​​(C is between A and B)):

We choose an arbitrary point Z that is not on the line is g and connects them with all three points. This gives the distances [AZ ], [ BZ ] and [ CZ]. Now we combine A with an arbitrary point S on the track [ BZ ], while the new line and the connecting line between Z and C intersect, this intersection we call G. joining us now B and [AZ ] so that the connection by the G runs. In [AZ ] we obtained by an intersection T. Pull we are now a straight line through T and S. The intersection of this line and the ground line g, which we call D, is the desired point = -1 induces the cross-ratio.

Projective geometry

In projective space the cross ratio can be calculated from the projective coordinates of the four collinear points, while it is independent of the particular choice of the coordinate system. Conversely projective coordinates can be regarded as cross ratios. → See projective coordinate system.

The cross ratio is a projective invariant of each figure, ie it retains the use of such a figure its value. This property can be regarded as characterizing feature of projective geometry. See also: Erlanger program. These relationships were known since ancient times and can be found eg in Pappus. They are the ultimate reason that the term double ratio was ever developed.

• Analytic Geometry