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- Lie_sphere_geometry abstract "Lie sphere geometry is a geometrical theory of planar or spatial geometry in which the fundamental concept is the circle or sphere. It was introduced by Sophus Lie in the nineteenth century. The main idea which leads to Lie sphere geometry is that lines (or planes) should be regarded as circles (or spheres) of infinite radius and that points in the plane (or space) should be regarded as circles (or spheres) of zero radius.The space of circles in the plane (or spheres in space), including points and lines (or planes) turns out to be a manifold known as the Lie quadric (a quadric hypersurface in projective space). Lie sphere geometry is the geometry of the Lie quadric and the Lie transformations which preserve it. This geometry can be difficult to visualize because Lie transformations do not preserve points in general: points can be transformed into circles (or spheres).To handle this, curves in the plane and surfaces in space are studied using their contact lifts, which are determined by their tangent spaces. This provides a natural realisation of the osculating circle to a curve, and the curvature spheres of a surface. It also allows for a natural treatment of Dupin cyclides and a conceptual solution of the problem of Apollonius.Lie sphere geometry can be defined in any dimension, but the case of the plane and 3-dimensional space are the most important. In the latter case, Lie noticed a remarkable similarity between the Lie quadric of spheres in 3-dimensions, and the space of lines in 3-dimensional projective space, which is also a quadric hypersurface in a 5-dimensional projective space, called the Plücker or Klein quadric. This similarity led Lie to his famous \"line-sphere correspondence\" between the space of lines and the space of spheres in 3-dimensional space.".
- Lie_sphere_geometry thumbnail Sophus_Lie.jpg?width=300.
- Lie_sphere_geometry wikiPageExternalLink lie_-_line_and_sphere_complexes.pdf.
- Lie_sphere_geometry wikiPageExternalLink helga_sopmath3_2.pdf.
- Lie_sphere_geometry wikiPageID "16646944".
- Lie_sphere_geometry wikiPageLength "27964".
- Lie_sphere_geometry wikiPageOutDegree "91".
- Lie_sphere_geometry wikiPageRevisionID "679965105".
- Lie_sphere_geometry wikiPageWikiLink American_Mathematical_Society.
- Lie_sphere_geometry wikiPageWikiLink Birkhäuser.
- Lie_sphere_geometry wikiPageWikiLink Category:Conformal_geometry.
- Lie_sphere_geometry wikiPageWikiLink Category:Differential_geometry.
- Lie_sphere_geometry wikiPageWikiLink Category:Incidence_geometry.
- Lie_sphere_geometry wikiPageWikiLink Celestial_sphere.
- Lie_sphere_geometry wikiPageWikiLink Centralizer_and_normalizer.
- Lie_sphere_geometry wikiPageWikiLink Circle.
- Lie_sphere_geometry wikiPageWikiLink Complex_number.
- Lie_sphere_geometry wikiPageWikiLink Contact_(mathematics).
- Lie_sphere_geometry wikiPageWikiLink Contact_geometry.
- Lie_sphere_geometry wikiPageWikiLink Cotangent_bundle.
- Lie_sphere_geometry wikiPageWikiLink Curvature_sphere.
- Lie_sphere_geometry wikiPageWikiLink Descartes_theorem.
- Lie_sphere_geometry wikiPageWikiLink Dupin_cyclide.
- Lie_sphere_geometry wikiPageWikiLink Edmond_Laguerre.
- Lie_sphere_geometry wikiPageWikiLink Euclidean_geometry.
- Lie_sphere_geometry wikiPageWikiLink Geometry.
- Lie_sphere_geometry wikiPageWikiLink Group_(mathematics).
- Lie_sphere_geometry wikiPageWikiLink Group_action.
- Lie_sphere_geometry wikiPageWikiLink Homogeneous_coordinates.
- Lie_sphere_geometry wikiPageWikiLink Hyperplane.
- Lie_sphere_geometry wikiPageWikiLink Hypersphere.
- Lie_sphere_geometry wikiPageWikiLink Immersion_(mathematics).
- Lie_sphere_geometry wikiPageWikiLink Indefinite_orthogonal_group.
- Lie_sphere_geometry wikiPageWikiLink Inversive_geometry.
- Lie_sphere_geometry wikiPageWikiLink Involution_(mathematics).
- Lie_sphere_geometry wikiPageWikiLink Isometry.
- Lie_sphere_geometry wikiPageWikiLink Isomorphism.
- Lie_sphere_geometry wikiPageWikiLink Klein_quadric.
- Lie_sphere_geometry wikiPageWikiLink Lagranges_identity.
- Lie_sphere_geometry wikiPageWikiLink Laguerre_geometry.
- Lie_sphere_geometry wikiPageWikiLink Lie_sphere_geometry.
- Lie_sphere_geometry wikiPageWikiLink Line_(geometry).
- Lie_sphere_geometry wikiPageWikiLink Linear_independence.
- Lie_sphere_geometry wikiPageWikiLink Linear_map.
- Lie_sphere_geometry wikiPageWikiLink Linear_subspace.
- Lie_sphere_geometry wikiPageWikiLink Manifold.
- Lie_sphere_geometry wikiPageWikiLink Minkowski_space.
- Lie_sphere_geometry wikiPageWikiLink Möbius_transformation.
- Lie_sphere_geometry wikiPageWikiLink Null_vector.
- Lie_sphere_geometry wikiPageWikiLink Orientation_(vector_space).
- Lie_sphere_geometry wikiPageWikiLink Origin_(mathematics).
- Lie_sphere_geometry wikiPageWikiLink Orthogonality.
- Lie_sphere_geometry wikiPageWikiLink Osculating_circle.
- Lie_sphere_geometry wikiPageWikiLink Pencil_(mathematics).
- Lie_sphere_geometry wikiPageWikiLink Plane_(geometry).
- Lie_sphere_geometry wikiPageWikiLink Plücker_coordinates.
- Lie_sphere_geometry wikiPageWikiLink Point_(geometry).
- Lie_sphere_geometry wikiPageWikiLink Point_at_infinity.
- Lie_sphere_geometry wikiPageWikiLink Problem_of_Apollonius.
- Lie_sphere_geometry wikiPageWikiLink Projective_line.
- Lie_sphere_geometry wikiPageWikiLink Projective_space.
- Lie_sphere_geometry wikiPageWikiLink Quadratic_form.
- Lie_sphere_geometry wikiPageWikiLink Quadric.
- Lie_sphere_geometry wikiPageWikiLink Radius.
- Lie_sphere_geometry wikiPageWikiLink Rocky_Mountain_Journal_of_Mathematics.
- Lie_sphere_geometry wikiPageWikiLink Sophus_Lie.
- Lie_sphere_geometry wikiPageWikiLink Spacetime.
- Lie_sphere_geometry wikiPageWikiLink Sphere.
- Lie_sphere_geometry wikiPageWikiLink Symmetric_bilinear_form.
- Lie_sphere_geometry wikiPageWikiLink Symmetry_in_mathematics.
- Lie_sphere_geometry wikiPageWikiLink Tangent.
- Lie_sphere_geometry wikiPageWikiLink Tangent_space.
- Lie_sphere_geometry wikiPageWikiLink Three-dimensional_space_(mathematics).
- Lie_sphere_geometry wikiPageWikiLink Walter_Benz.
- Lie_sphere_geometry wikiPageWikiLink File:Apollonius_all_solutions.png.
- Lie_sphere_geometry wikiPageWikiLink File:Cyclide.png.
- Lie_sphere_geometry wikiPageWikiLink File:Ruled_hyperboloid.jpg.
- Lie_sphere_geometry wikiPageWikiLink File:Sophus_Lie.jpg.
- Lie_sphere_geometry wikiPageWikiLinkText "Lie sphere geometry".
- Lie_sphere_geometry wikiPageWikiLinkText "Lie sphere geometry#The Lie quadric".
- Lie_sphere_geometry wikiPageWikiLinkText "Lie-equivalent".
- Lie_sphere_geometry wikiPageWikiLinkText "geometry of circles and spheres".
- Lie_sphere_geometry wikiPageUsesTemplate Template:Citation.
- Lie_sphere_geometry wikiPageUsesTemplate Template:Reflist.
- Lie_sphere_geometry subject Category:Conformal_geometry.
- Lie_sphere_geometry subject Category:Differential_geometry.
- Lie_sphere_geometry subject Category:Incidence_geometry.
- Lie_sphere_geometry hypernym Theory.
- Lie_sphere_geometry type Work.
- Lie_sphere_geometry type Combinatoric.
- Lie_sphere_geometry type Physic.
- Lie_sphere_geometry comment "Lie sphere geometry is a geometrical theory of planar or spatial geometry in which the fundamental concept is the circle or sphere. It was introduced by Sophus Lie in the nineteenth century.".
- Lie_sphere_geometry label "Lie sphere geometry".
- Lie_sphere_geometry sameAs Q17103254.
- Lie_sphere_geometry sameAs Geometria_de_lesfera_de_Lie.
- Lie_sphere_geometry sameAs m.03yjgbm.
- Lie_sphere_geometry sameAs Q17103254.
- Lie_sphere_geometry wasDerivedFrom Lie_sphere_geometry?oldid=679965105.
- Lie_sphere_geometry depiction Sophus_Lie.jpg.
- Lie_sphere_geometry isPrimaryTopicOf Lie_sphere_geometry.