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,,, 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.
The point is the harmonic conjugate of with respect to and precisely if the cross-ratio of the quadruple is, 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.
Terminology and history
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. Chasles coined the French term rapport anharmonique in 1837. German geometers call it das Doppelverhältnis .
Carl von Staudt was unsatisfied with past definitions of the cross-ratio relying on algebraic manipulation of Euclidean distances rather than being based purely on synthetic projective geometry concepts. In 1847, von Staudt demonstrated that the algebraic structure is implicit in projective geometry, by creating an algebra based on construction of the projective harmonic conjugate, which he called a throw : given three points on a line, the harmonic conjugate is a fourth point that makes the cross ratio equal to. 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.
Definition
If,,, and are four points on an oriented affine line, their cross ratio is:with the notation defined to mean the signed ratio of the displacement from to to the displacement from to. For collinear displacements this is a dimensionless quantity.
If the displacements themselves are taken to be signed real numbers, then the cross ratio between points can be written
If is the projectively extended real line, the cross-ratio of four distinct numbers in is given by
When one of is the point at infinity this reduces to e.g.
The same formulas can be applied to four distinct complex numbers or, more generally, to elements of any field, and can also be projectively extended as above to the case when one of them is
The cross ratio can for example be defined for pencils of lines, circles, or conics. For instance, the cross-ratio of coaxial circles can be defined in numerous equivalent ways:
- Let be a point of intersection of the circles on their radical axis, if it exists. Then the cross-ratio of the circles can be defined as the cross-ratio of the tangents to the circles through.
- More generally, given any point in the plane, the polars of this point with respect to those circles are concurrent and their cross-ratio doesn't depend on the chosen point.
- By taking the lines orthogonal to the tangents at and projecting on the line on which lies the circle centers, we deduce it is equal to the cross-ratio of the circle centers.
- It can be proven through an inversion that the cross-ratio of these circles can be equivalently defined as the cross-ratio of the second points of intersection different than of a circle that passes through.
Properties
where describes the ratio with which the point divides the line segment, and describes the ratio with which the point divides that same line segment. The cross ratio then appears as a ratio of ratios, describing how the two points and are situated with respect to the line segment. As long as the points,,, and are distinct, the cross ratio will be a non-zero real number. We can easily deduce that
- if and only if one of the points or lies between the points and and the other does not
Six cross-ratios
See Anharmonic group below.
Projective geometry
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 points lie on a straight line in then their cross-ratio is a well-defined quantity, because any choice of the origin and even of the scale on the line will yield the same value of the cross-ratio.
Furthermore, let be four distinct lines in the plane passing through the same point. Then any line not passing through intersects these lines in four distinct points . It turns out that the cross-ratio of these points does not depend on the choice of a line, and hence it is an invariant of the 4-tuple of lines
This can be understood as follows: if and are two lines not passing through then the perspective transformation from to with the center is a projective transformation that takes the quadruple of points on into the quadruple of points on.
Therefore, the invariance of the cross-ratio under projective automorphisms of the line implies the independence of the cross-ratio of the four collinear points on the lines from the choice of the line that contains them.
Definition in homogeneous coordinates
If four collinear points are represented in homogeneous coordinates by vectors such that and, then their cross-ratio is.Role in non-Euclidean geometry
and Felix Klein found an application of the cross-ratio to non-Euclidean geometry. Given a nonsingular conic in the real projective plane, its stabilizer in the projective group acts transitively on the points in the interior of. However, there is an invariant for the action of on pairs of points. In fact, every such invariant is expressible as a function of the appropriate cross-ratio.Hyperbolic geometry
Explicitly, let the conic be the unit circle. For any two points and, inside the unit circle. If the line connecting them intersects the circle in two points, and and the points are, in order,. Then the hyperbolic distance between and in the Cayley–Klein model of the hyperbolic plane can be expressed as. Since the cross-ratio is invariant under projective transformations, it follows that the hyperbolic distance is invariant under the projective transformations that preserve the conic.
Conversely, the group acts transitively on the set of pairs of points in the unit disk at a fixed hyperbolic distance.
Later, partly through the influence of Henri Poincaré, the cross ratio of four complex numbers on a circle was used for hyperbolic metrics. Being on a circle means the four points are the image of four real points under a Möbius transformation, and hence the cross ratio is a real number. The Poincaré half-plane model and Poincaré disk model are two models of hyperbolic geometry in the complex projective line.
These models are instances of Cayley–Klein metrics.
Anharmonic group and Klein four-group
The cross-ratio may be defined by any of these four expressions:These differ by the following permutations of the variables :
We may consider the permutations of the four variables as an action of the symmetric group on functions of the four variables. Since the above four permutations leave the cross ratio unaltered, they form the stabilizer of the cross-ratio under this action, and this induces an effective action of the quotient group on the orbit of the cross-ratio. The four permutations in provide a realization of the Klein four-group in, and the quotient is isomorphic to the symmetric group.
Thus, the other permutations of the four variables alter the cross-ratio to give the following six values, which are the orbit of the six-element group :
File:Symmetries of the anharmonic group.png|thumb|300px|
The stabilizer of is isomorphic to the rotation group of the trigonal dihedron, the dihedral group. It is convenient to visualize this by a Möbius transformation mapping the real axis to the complex unit circle, with equally spaced.
Considering as the vertices of the dihedron, the other fixed points of
the -cycles are the points which under are opposite each vertex on the Riemann sphere, at the midpoint of the opposite edge. Each -cycles is a half-turn rotation of the Riemann sphere exchanging the hemispheres.
The fixed points of the -cycles are, corresponding under to the poles of the sphere: is the origin and is the point at infinity. Each -cycle is a turn rotation about their axis, and they are exchanged by the -cycles.
As functions of these are examples of Möbius transformations, which under composition of functions form the Mobius group. The six transformations form a subgroup known as the anharmonic group, again isomorphic to. They are the torsion elements in. Namely,,, and are of order with respective fixed points and . Meanwhile, the elements
and are of order in, and each fixes both values of the "most symmetric" cross-ratio. The order elements exchange these two elements, and thus the action of the anharmonic group on gives the quotient map of symmetric groups.
Further, the fixed points of the individual -cycles are, respectively, and and this set is also preserved and permuted by the -cycles. Geometrically, this can be visualized as the rotation group of the trigonal dihedron, which is isomorphic to the dihedral group of the triangle, as illustrated at right. Algebraically, this corresponds to the action of on the -cycles by conjugation and realizes the isomorphism with the group of inner automorphisms,
The anharmonic group is generated by and Its action on gives an isomorphism with. It may also be realised as the six Möbius transformations mentioned, which yields a projective representation of over any field, and is always faithful/injective. Over the field with two elements, the projective line only has three points, so this representation is an isomorphism, and is the exceptional isomorphism. In characteristic, this stabilizes the point, which corresponds to the orbit of the harmonic cross-ratio being only a single point, since. Over the field with three elements, the projective line has only 4 points and, and thus the representation is exactly the stabilizer of the harmonic cross-ratio, yielding an embedding equals the stabilizer of the point.