Cross-ratio

In mathematics, the cross-ratio, also called double ratio and anharmonic ratio, is a numerical invariant of an ordered 4-tuple of distinct points on a line and of an ordered 4-tuple of concurrent lines in a plane. It had been defined in deep antiquity, possibly already by Euclid, and was considered by Pappus, who remarked its key invariance property. It was extensively studied in the 19th century.

The cross-ratio is preserved by the fractional linear transformations. It plays a prominent role in projective geometry because it is the only projective invariant of an ordered quadruple of points on a projective line.

Definition

The cross-ratio of a 4-tuple of distinct points on the real line with coordinates z₁, z₂, z₃, z₄ is a non-zero real number given by the formula

(z_{1},z_{2};z_{3},z_{4})={\frac {(z_{1}-z_{3})(z_{2}-z_{4})}{(z_{2}-z_{3})(z_{1}-z_{4})}}.

It may be more suggestively rewritten as a "double ratio" of two "simple ratios" of triples of points:

(z_{1},z_{2};z_{3},z_{4})={\frac {z_{1}-z_{3}}{z_{2}-z_{3}}}:{\frac {z_{1}-z_{4}}{z_{2}-z_{4}}}.

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. Thus the cross-ratio can be viewed as an invariant of points on a projective line.

In geometry, if A, B, C and D are colinear points, then the cross ratio is defined similarly as

(A,B;C,D)={\frac {AC\cdot BD}{BC\cdot AD}}.

where each of the distances is signed according to a fixed orientation of the line.

Terminology and history

The English term "cross-ratio" was introduced by Clifford. German geometers of the 19th century called it das Doppelverhältnis (Ger: double ratio) and French mathematicians after Chasles used the term le rapport anharmonique (Fr: anharmonic ratio).

In a geometric context, the notion of a cross-ratio goes back to antiquity. A theorem on the anharmonic ratio of lines appeared in the work of Pappus, but Michel Chasles, who devoted considerable efforts to reconstructing lost works of Euclid, asserted that it had earlier appeared in his book Porisms.

Projective geometry

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 L in R² 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 {L_i, 1 ≤ i ≤ 4}, be four distinct lines in the plane passing through the same point Q. Then any line L not passing through Q intersects these lines in four distinct points P_i (if L is parallel to L_i then the corresponding intersection point is "at infinity"). It turns out that the cross-ratio of these points (taken in a fixed order) does not depend on the choice of a line L, and hence it is an invariant of the 4-tuple of lines {L_i}. This can be understood as follows: if L and L′ are two lines not passing through Q then the perspective transformation from L to L′ with the center Q is a projective transformation that takes the quadruple {P_i} of points on L into the quadruple {P_i′} of points on L′. Therefore, the invariance of the cross-ratio under projective automorphisms of the line implies (in fact, is equivalent to) the independence of the cross-ratio of the four collinear points {P_i} on the lines {L_i} from the choice of the line that contains them.

Role in non-Euclidean geometry

Arthur Cayley and Felix Klein found an application of the cross-ratio to non-Euclidean geometry. Given a nonsingular conic C in the real projective plane, its stabilizer G in the projective group PGL(3,R) acts transitively on the points in the interior of C. However, there is an invariant for the action of G on the pairs of points. In fact, every such invariant is expressible as a function of the appropriate cross ratio.

Explicitly, let the conic be the unit circle. For any two points in the unit disk, p, q, the line connecting them intersects the circle in two points, a and b. The points are, in order, a,p,q,b. Then the distance between p and q in the Cayley–Klein model of the plane hyperbolic geometry can be expressed as

d(p,q)={\frac {1}{2}}\log {\frac {|q-a||b-p|}{|p-a||b-q|}}

(the factor one half is needed to make the curvature −1). 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 C. Conversely, the group G acts transitively on the set of pairs of points (p,q) in the unit disk at a fixed hyperbolic distance.

Symmetry

There are different definitions of the cross-ratio used in the literature. However, they all differ from each other by a some possible permutation of the coordinates. In general, there are six possible different values the cross-ratio can take depending on the order in which the points z_i are given. Since there are 24 possible permutations of the four coordinates, some permutations must leave the cross-ratio unaltered. In fact, exchanging any two pairs of coordinates preserves the cross-ratio:

(z_{1},z_{2};z_{3},z_{4})=(z_{2},z_{1};z_{4},z_{3})=(z_{3},z_{4};z_{1},z_{2})=(z_{4},z_{3};z_{2},z_{1}).\,

Using these symmetries, there can then be 6 possible values of the cross-ratio, depending on the order in which the points are given. These are:

$(z_{1},z_{2};z_{3},z_{4})=\lambda \,$		$(z_{1},z_{2};z_{4},z_{3})={1 \over \lambda }$
$(z_{1},z_{3};z_{4},z_{2})={1 \over {1-\lambda }}$		$(z_{1},z_{3};z_{2},z_{4})=1-\lambda \,$
$(z_{1},z_{4};z_{3},z_{2})={\lambda \over {\lambda -1}}$		$(z_{1},z_{4};z_{2},z_{3})={{\lambda -1} \over \lambda }$

In the language of group theory, the symmetric group S₄ acts on the cross-ratio by permuting coordinates. The kernel of this action is the Klein four-group (this is the group which preserves the cross-ratio). The effective symmetry group is then the quotient group which is isomorphic to S₃.

For certain values of λ there will be an enhanced symmetry and therefore fewer than six possible values for the cross-ratio. These values of λ correspond to fixed points of the action of S₃ on the Riemann sphere (given by the above six functions); or, equivalently, those points with a non-trivial stabilizer in this permutation group.

The first set of fixed points is {0, 1, ∞}. However, the cross-ratio can never take on these values if the points {z_i} are all distinct. These values are limit values as one pair of coordinates approach each other:

(z,z_{2};z,z_{4})=(z_{1},z;z_{3},z)=0\,

(z,z;z_{3},z_{4})=(z_{1},z_{2};z,z)=1\,

(z,z_{2};z_{3},z)=(z_{1},z;z,z_{4})=\infty .

The second set of fixed points is {−1, 1/2, 2}. This situation is what is classically called the harmonic cross-ratio, and arises in projective harmonic conjugates.

The most symmetric cross-ratio occurs when $\lambda =e^{\pm i\pi /3}$ . These are then the only two possible values of the cross-ratio.

Transformational approach

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 transformation are known as Möbius transformations. A general Möbius transformation has the form

f(z)={\frac {az+b}{cz+d}}\;,\quad {\mbox{where }}a,b,c,d\in {\mathbb {C}}{\mbox{ and }}ad-bc\neq 0.

These transformations form a group acting on the Riemann sphere.

The projective invariance of the cross-ratio means that

(f(z_{1}),f(z_{2});f(z_{3}),f(z_{4}))=(z_{1},z_{2};z_{3},z_{4}).\

The cross-ratio is real if and only if the four points are either collinear or concyclic, reflecting the fact that every Möbius transformation maps generalized circles to generalized circles.

The action of the Möbius group is simply transitive on the set of triples of distinct points of the Riemann sphere: given any ordered triple of distinct points, (z₂,z₃,z₄), there is a unique Möbius transformation f(z) that maps it to the triple (1,0,∞). This transformation can be conveniently described using the cross-ratio: since (z,z₂,z₃,z₄) must equal (f(z),1;0,∞) which in turn equals f(z), we obtain

f(z)=(z,z_{2};z_{3},z_{4}).\,

Differential-geometric point of view

The theory takes on a differential calculus aspect as the four points are brought into proximity. This leads to the theory of the Schwarzian derivative, and more generally of projective connections.

External links

MathPages - Kevin Brown explains the cross-ratio in his article about Pascal's Mystic Hexagram
Java Applet demonstrating the invariance of the cross ratio under a bilinear transformation
Cross-Ratio at cut-the-knot