Real projective line

In geometry, a real projective line is an extension of the usual concept of line that has been historically introduced to solve a problem set by visual perspective: two parallel lines do not intersect but seem to intersect "at infinity". For solving this problem, points at infinity have been introduced, in such a way that in a real projective plane, two distinct projective lines meet in exactly one point. The set of these points at infinity, the "horizon" of the visual perspective in the plane, is a real projective line. It is the circle of directions emanating from an observer situated at any point, with opposite points identified. A model of the real projective line is the projectively extended real line. Drawing a line to represent the horizon in visual perspective, an additional point at infinity is added to represent the collection of lines parallel to the horizon.

Formally, the real projective line is defined as the space of all one-dimensional linear subspaces of a two-dimensional vector space over the reals. Accordingly, the general linear group of 2×2 invertible matrices acts on the real projective line. Since the center acts trivially, the projective linear group, PGL(2, R), also acts on the projective line. These are the geometric transformations of the projective line. When the projective line is represented as a real line with point at infinity, the elements of the projective linear group act as fractional linear transformations. These transformations of the real projective line are called homographies.

Topologically, the real projective line is homeomorphic to the circle. The real projective line is the boundary of the hyperbolic plane. Every isometry of the hyperbolic plane induces a unique geometric transformation of the boundary, and vice versa. Furthermore, every harmonic function on the hyperbolic plane is given as a Poisson integral of a distribution on the projective line, in a manner that is compatible with the action of the isometry group. The topological circle has many compatible projective structures on it; the space of such structures is the (infinite dimensional) universal Teichmüller space. The complex analog of the real projective line is the complex projective line; that is, the Riemann sphere.

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