Cross-ratio

In geometry, the cross-ratio, also called the double ratio and anharmonic ratio, is a number associated with a list of four collinear points, particularly points on a projective line. Given four points A, B, C and D on a line, their cross ratio is defined as In geometry, the cross-ratio, also called the double ratio and anharmonic ratio, is a number associated with a list of four collinear points, particularly points on a projective line. Given four points A, B, C and D on a line, their cross ratio is defined as where an orientation of the line determines the sign of each distance and the distance is measured as projected into Euclidean space. (If one of the four points is the line's point at infinity, then the two distances involving that point are dropped from the formula.)The point D is the harmonic conjugate of C with respect to A and B precisely if the cross-ratio of the quadruple is −1, called the harmonic ratio. The cross-ratio can therefore be regarded as measuring the quadruple's deviation from this ratio; hence the name anharmonic ratio. The cross-ratio is preserved by linear fractional transformations. It is essentially the only projective invariant of a quadruple of collinear points; this underlies its importance for projective geometry. The cross-ratio had been defined in deep antiquity, possibly already by Euclid, and was considered by Pappus, who noted its key invariance property. It was extensively studied in the 19th century. Variants of this concept exist for a quadruple of concurrent lines on the projective plane and a quadruple of points on the Riemann sphere.In the Cayley–Klein model of hyperbolic geometry, the distance between points is expressed in terms of a certain cross-ratio. 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 made a translation of the original by Pappus, then wrote a commentary on how the lemmas of Pappus relate to modern terminology. Modern use of the cross ratio in projective geometry began with Lazare Carnot in 1803 with his book Géométrie de Position. The term used was le rapport anharmonique (Fr: anharmonic ratio). German geometers call it das Doppelverhältnis (Ger: double ratio). Given three points on a line, a fourth point that makes the cross ratio equal to minus one is called the projective harmonic conjugate. In 1847 Carl von Staudt called the construction of the fourth point a throw (Wurf), and used the construction to exhibit arithmetic implicit in geometry. His Algebra of Throws provides an approach to numerical propositions, usually taken as axioms, but proven in projective geometry. The English term 'cross-ratio' was introduced in 1878 by William Kingdon Clifford.