Cross-Ratio

$$[a,b,c,d]\equiv {(a-b)(c-d)\over (a-d)(c-b)}.$$ (1)

For a Möbius Transformation $f$,

$$[a,b,c,d]= [f(a),f(b),f(c),f(d)].$$ (2)

There are six different values which the cross-ratio may take, depending on the order in which the points are chosen. Let $\lambda \equiv [a,b,c,d]$. Possible values of the cross-ratio are then $\lambda$, $1-\lambda$, $1/\lambda$, $(\lambda-1)/\lambda$, $1/(1-\lambda)$, and $\lambda/(\lambda-1)$.

Given lines $a$, $b$, $c$, and $d$ which intersect in a point $O$, let the lines be cut by a line $l$, and denote the points of intersection of $l$ with each line by $A$, $B$, $C$, and $D$. Let the distance between points $A$ and $B$ be denoted $AB$, etc. Then the cross-ratio

$$[AB,CD]\equiv {(AB)(CD)\over(BC)(AD)}$$ (3)

is the same for any position of the $l$ (Coxeter and Greitzer 1967). Note that the definitions $(AB/AD)/(BC/CD)$ and $(CA/CB)/(DA/DB)$ are used instead by Kline (1990) and Courant and Robbins (1966), respectively. The identity

$$[AD,BC]+[AB,DC]=1$$ (4)

holds Iff $AC // BD$, 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.