Cross-Ratio
 | (1) |
, | (2) |
. 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 thepoints of intersection of with each line by 
, 
, 
, and 
. Let the distance between points and bedenoted 
, etc. Then the cross-ratio
 | (3) |
(Coxeter and Greitzer 1967). Note that the definitions 
and 
are used instead by Kline (1990) and Courant and Robbins (1966), respectively.The identity | (4) |
, 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).
See also Möbius Transformation, Separation
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.
- Stirling Number of the Second Kind
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