Cross-ratio - Role in Non-Euclidean Geometry

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_C in the projective group G = PGL(3,R) acts transitively on the points in the interior of C. However, there is an invariant for the action of G_C on 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

(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.