## Cross-Ratio

 (1)

For a Möbius Transformation ,
 (2)

There are six different values which the cross-ratio may take, depending on the order in which the points are chosen. Let . Possible values of the cross-ratio are then , , , , , and .

Given lines , , , and which intersect in a point , let the lines be cut by a line , and denote the points of intersection of with each line by , , , and . Let the distance between points and be denoted , etc. Then the cross-ratio

 (3)

is the same for any position of the (Coxeter and Greitzer 1967). Note that the definitions and are used instead by Kline (1990) and Courant and Robbins (1966), respectively. The identity
 (4)

holds Iff , where denotes Separation.

The cross-ratio of four points on a radial line of an Inversion Circle is preserved under Inversion (Ogilvy 1990, p. 40).

References

Courant, R. and Robbins, H. What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, 1996.

Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 107-108, 1967.

Kline, M. Mathematical Thought from Ancient to Modern Times, Vol. 1. Oxford, England: Oxford University Press, 1990.

Ogilvy, C. S. Excursions in Geometry. New York: Dover, pp. 39-41, 1990.