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 P(R) is defined as the space of all one-dimensional linear subspaces of a two-dimensional vector space over the reals. The automorphisms of the real projective line are constructed with 2 × 2 real matrices. A matrix is required to be non-singular, and following the identification of proportional projective coordinates, proportional matrices (having identical actions on the real projective line) determine the same automorphism of P(R). Such an automorphism is sometimes called a homography of the projective line. With due regard for the point at infinity, an automorphism may be called a linear fractional transformation. The automorphisms form the projective linear group PGL(2,R). 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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