Pappus of Alexandria made implicit use of concepts equivalent to the cross-ratio in his Collection: Book VII. Early users of Pappus included Isaac Newton, Michel Chasles, and Robert Simson. In 1986 Alexander Jones … See more

The cross-ratio is a projective invariant in the sense that it is preserved by the projective transformations of a projective line.
In particular, if four … See more

Arthur Cayley and Felix Klein found an application of the cross-ratio to non-Euclidean geometry. Given a nonsingular conic See more

The cross-ratio is invariant under the projective transformations of the line. In the case of a complex projective line, or the Riemann sphere, these transformations are … See more

The cross ratio of the four collinear points A, B, C, and D can be written as
$${\displaystyle (A,B;C,D)={\frac {AC:CB}{AD:DB}}}$$
where See more

If four collinear points are represented in homogeneous coordinates by vectors $${\displaystyle \alpha ,\beta ,\gamma ,\delta }$$ such that See more

The cross-ratio may be defined by any of these four expressions:
$${\displaystyle (A,B;C,D)=(B,A;D,C)=(C,D;A,B)=(D,C;B,A).}$$
These differ by the following permutations of the variables (in See more