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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 different 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, , 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. ==Definition== 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, . Define on the binary relation to hold when there exists a nonzero real number such that . 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 is the set of all equivalence classes. Each equivalence class is considered as a single point, or, in other words, a ''point'' is defined as being an equivalence class. If one chooses a basis of , this amounts (by identifying a vector with its coordinates vector) to identify with the direct product , and the equivalence relation becomes if there exists a nonzero real number such that . In this case, the projective line is preferably denoted or . The equivalence class of the pair is traditionally denoted , the colon in the notation recalling that, if , the ratio is the same for all elements of the equivalence class. If a point is the equivalence class one says that is a pair of projective coordinates of .〔The argument used to construct can also be used with any field ''K'' and any dimension to construct the projective space .〕 As is defined through an equivalence relation, the canonical projection from to defines a topology (the quotient topology) and a differential structure on the projective line. However, the fact that equivalence classes are not finite induces some difficulties for defining the differential structure. These are solved by considering as an Euclidean vector space. The circle of the unit vectors is, in the case of , the set of the vectors whose coordinates satisfy . This circle intersects each equivalence classes in exactly two opposite points. Therefore, the projective line may be considered a the quotient space of the circle by the equivalence relation such that if and only if either or . 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Real projective line」の詳細全文を読む スポンサード リンク
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