If , , ,
and are points in the extended
complex plane ,
their cross ratio, also called the cross-ratio (Courant and Robbins 1996, p. 172;
Durell 1928, p. 73), anharmonic ratio, and anharmonic section (Casey 1888),
is defined by
(1)
Here if ,
the result is infinity, and if one of , ,
, or is infinity, then the two terms on the right containing it
are cancelled.
The function
is the unique linear fractional transformation
which takes
to 0,
to 1, and
to infinity. Moreover, is real if and only if the four points lie on a straight
line or a generalized circle.
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 four collinear points , ,
, and , let the distance between points and
be denoted ,
etc. Then the cross ratio can be defined by
(3)
The notation
is sometimes also used (Coxeter and Greitzer 1967, p. 107).
There are a number of different notational and definition conventions for the cross ratio. For example, the definitions and are used by Kline (1990) and Courant
and Robbins (1966), respectively (Coxeter and Greitzer 1967, p. 107).
The cross ratio can also be defined for any four coplanar points. It is preserved by any inversion (cf. Ogilvy 1990, p. 40) whose pole
is different from any of the given four points (where the last restriction is necessary
only to avoid working with infinities).
The following table summarized the cross ratios for some geometric configurations.