Definition
The cross-ratio of a 4-tuple of distinct points on the real line with coordinates z1, z2, z3, z4 is given by
It can also be written as a "double ratio" of two division ratios of triples of points:
The same formulas can be applied to four different complex numbers or, more generally, to elements of any field and can also be extended to the case when one of them is the symbol ∞, by removing the corresponding two differences from the formula. The formula shows that cross-ratio is a function of four points, generally four numbers taken from a field. The generalization using elements of a mathematical ring requires methods of inversive ring geometry.
In geometry, if A, B, C and D are collinear points, then the cross ratio is defined similarly as
where each of the distances is signed according to a fixed orientation of the line.
Read more about this topic: Cross-ratio
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