Matches in DBpedia 2015-10 for { <http://dbpedia.org/resource/Real_projective_line> ?p ?o }
- Real_projective_line abstract "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, 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.".
- Real_projective_line thumbnail Real_projective_line.svg?width=300.
- Real_projective_line wikiPageExternalLink projective.pdf.
- Real_projective_line wikiPageExternalLink chapter_two.html.
- Real_projective_line wikiPageID "46379102".
- Real_projective_line wikiPageLength "9798".
- Real_projective_line wikiPageOutDegree "68".
- Real_projective_line wikiPageRevisionID "675483933".
- Real_projective_line wikiPageWikiLink 2_×_2_real_matrices.
- Real_projective_line wikiPageWikiLink Analytic_function.
- Real_projective_line wikiPageWikiLink Analytic_manifold.
- Real_projective_line wikiPageWikiLink Atlas_(topology).
- Real_projective_line wikiPageWikiLink Binary_relation.
- Real_projective_line wikiPageWikiLink Canonical_projection.
- Real_projective_line wikiPageWikiLink Category:Manifolds.
- Real_projective_line wikiPageWikiLink Category:Projective_geometry.
- Real_projective_line wikiPageWikiLink Central_projection.
- Real_projective_line wikiPageWikiLink Chart_(topology).
- Real_projective_line wikiPageWikiLink Circle.
- Real_projective_line wikiPageWikiLink Complete_metric_space.
- Real_projective_line wikiPageWikiLink Complete_space.
- Real_projective_line wikiPageWikiLink Complex_projective_line.
- Real_projective_line wikiPageWikiLink Coordinate_vector.
- Real_projective_line wikiPageWikiLink Coordinates_vector.
- Real_projective_line wikiPageWikiLink Cyclic_order.
- Real_projective_line wikiPageWikiLink Differentiable_function.
- Real_projective_line wikiPageWikiLink Differentiable_manifold.
- Real_projective_line wikiPageWikiLink Differential_structure.
- Real_projective_line wikiPageWikiLink Embedding.
- Real_projective_line wikiPageWikiLink Equivalence_class.
- Real_projective_line wikiPageWikiLink Equivalence_relation.
- Real_projective_line wikiPageWikiLink Euclidean_space.
- Real_projective_line wikiPageWikiLink Euclidean_vector_space.
- Real_projective_line wikiPageWikiLink Fractional_linear_transformation.
- Real_projective_line wikiPageWikiLink General_linear_group.
- Real_projective_line wikiPageWikiLink Geometric_transformation.
- Real_projective_line wikiPageWikiLink Geometry.
- Real_projective_line wikiPageWikiLink Harmonic_function.
- Real_projective_line wikiPageWikiLink Homeomorphic.
- Real_projective_line wikiPageWikiLink Homeomorphism.
- Real_projective_line wikiPageWikiLink Homogeneous_coordinates.
- Real_projective_line wikiPageWikiLink Homography.
- Real_projective_line wikiPageWikiLink Hyperbolic_geometry.
- Real_projective_line wikiPageWikiLink Hyperbolic_plane.
- Real_projective_line wikiPageWikiLink Hyperplane_at_infinity.
- Real_projective_line wikiPageWikiLink Inverse_function.
- Real_projective_line wikiPageWikiLink Line_(geometry).
- Real_projective_line wikiPageWikiLink Linear_fractional_transformation.
- Real_projective_line wikiPageWikiLink Manifold.
- Real_projective_line wikiPageWikiLink Mathematical_structure.
- Real_projective_line wikiPageWikiLink Modular_group.
- Real_projective_line wikiPageWikiLink Multiplicative_inverse.
- Real_projective_line wikiPageWikiLink Möbius_group.
- Real_projective_line wikiPageWikiLink Möbius_transformation.
- Real_projective_line wikiPageWikiLink New_York_University.
- Real_projective_line wikiPageWikiLink Northwestern_University.
- Real_projective_line wikiPageWikiLink Parallel_(geometry).
- Real_projective_line wikiPageWikiLink Parallel_lines.
- Real_projective_line wikiPageWikiLink Parallel_projection.
- Real_projective_line wikiPageWikiLink Perspective_(visual).
- Real_projective_line wikiPageWikiLink Point_at_infinity.
- Real_projective_line wikiPageWikiLink Points_at_infinity.
- Real_projective_line wikiPageWikiLink Poisson_integral.
- Real_projective_line wikiPageWikiLink Poisson_kernel.
- Real_projective_line wikiPageWikiLink Projection_(mathematics).
- Real_projective_line wikiPageWikiLink Projective_coordinates.
- Real_projective_line wikiPageWikiLink Projective_harmonic_conjugate.
- Real_projective_line wikiPageWikiLink Projective_harmonic_conjugates.
- Real_projective_line wikiPageWikiLink Projective_linear_group.
- Real_projective_line wikiPageWikiLink Projective_range.
- Real_projective_line wikiPageWikiLink Projectively_extended_real_line.
- Real_projective_line wikiPageWikiLink Quotient_space_(topology).
- Real_projective_line wikiPageWikiLink Quotient_topology.
- Real_projective_line wikiPageWikiLink Ratio.
- Real_projective_line wikiPageWikiLink Real_line.
- Real_projective_line wikiPageWikiLink Real_projective_plane.
- Real_projective_line wikiPageWikiLink Real_vector_space.
- Real_projective_line wikiPageWikiLink Riemann_sphere.
- Real_projective_line wikiPageWikiLink Separation_relation.
- Real_projective_line wikiPageWikiLink Slope.
- Real_projective_line wikiPageWikiLink Total_order.
- Real_projective_line wikiPageWikiLink Totally_ordered.
- Real_projective_line wikiPageWikiLink Transition_map.
- Real_projective_line wikiPageWikiLink Unit_vector.
- Real_projective_line wikiPageWikiLink Universal_Teichmüller_space.
- Real_projective_line wikiPageWikiLink Vector_space.
- Real_projective_line wikiPageWikiLink File:Real_projective_line.svg.
- Real_projective_line wikiPageWikiLinkText "Real projective line".
- Real_projective_line wikiPageWikiLinkText "example of a hyperbolic involution".
- Real_projective_line wikiPageWikiLinkText "projective line".
- Real_projective_line wikiPageWikiLinkText "real projective line".
- Real_projective_line wikiPageWikiLinkText "∞".
- Real_projective_line hasPhotoCollection Real_projective_line.
- Real_projective_line wikiPageUsesTemplate Template:Math.
- Real_projective_line wikiPageUsesTemplate Template:Merge.
- Real_projective_line wikiPageUsesTemplate Template:Reflist.
- Real_projective_line wikiPageUsesTemplate Template:See_also.
- Real_projective_line subject Category:Manifolds.
- Real_projective_line subject Category:Projective_geometry.
- Real_projective_line hypernym Extension.