The cross-ratio
In modeling visual perception, the Renaissance painters brought forth a new geometry to represent depth. Nothing seemed to be preserved, except a strange relation linking four collinear points, the cross-ratio, which however, since Antiquity, had been glimpsed by Menelaus and Pappus. This invariant, which emerges from the concurrence of four lines, allows geometric constructions and elegant proofs, even in a non-Euclidean setting. It offers a different perspective on the plane, by abolishing the reign of the notions of angle and distance.
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The theorems of Menelaus and Ceva
Menelaus's and Ceva's theorems, two classics of plane geometry, are similar in form. This resemblance becomes clearer when the notion of cross-ratio is introduced.
A geometric tool of unparalleled power
The harmonic range, like the more general notion of cross-ratio, has proved essential to geometric reasoning, particularly when dealing with cocyclicity—the property of points in the plane lying on the same circle—or pencils of lines.
A cross-ratio from another world
During the 19th century, the search for quantities invariant under a particular group of transformations became the main focus of the various branches of geometry. The cross-ratio, a fundamental invariant of projective geometry, also appears in non-Euclidean geometries and their models.
An invariant under central projection
The need to model visual perception gave rise to a new geometry. Renaissance painters felt compelled to study it closely in order to depict depth. Lengths, angles: nothing seemed to be preserved, apart from a curious relation linking four collinear points…