Real projective line

In geometry, a real projective line is a projective line over the real numbers. It is an extension of the usual concept of a 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 set of directions emanating from an observer situated at any point, with opposite directions identified. An example of a real projective line is the projectively extended real line, which is often called the projective line. Formally, a real projective line P(R) is defined as the set of all one-dimensional linear subspaces of a two-dimensional vector space over the reals. The automorphisms of a real projective line are called projective transformations, homographies, or linear fractional transformations. They form the projective linear group PGL(2, R). Each element of PGL(2, R) can be defined by a nonsingular 2×2 real matrix, and two matrices define the same element of PGL(2, R) if one is the product of the other and a nonzero real number. Topologically, real projective lines are homeomorphic to circles. The complex analog of a real projective line is a complex projective line, also called a Riemann sphere. The points of the real projective line are usually defined as equivalence classes of an equivalence relation. The starting point is a real vector space of dimension 2, V. Define on V ∖ 0 the binary relation v ~ w to hold when there exists a nonzero real number t such that v = tw. The definition of a vector space implies almost immediately that this is an equivalence relation. The equivalence classes are the vector lines from which the zero vector has been removed. The real projective line P(V) is the set of all equivalence classes.