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- Projective_geometry abstract "In mathematics, projective geometry is the study of geometric properties that are invariant under projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts. The basic intuitions are that projective space has more points than Euclidean space, in a given dimension, and that geometric transformations are permitted that move the extra points (called "points at infinity") to traditional points, and vice versa.Properties meaningful in projective geometry are respected by this new idea of transformation, which is more radical in its effects than expressible by a transformation matrix and translations (the affine transformations). The first issue for geometers is what kind of geometric language is adequate to the novel situation? It is not possible to talk about angles in projective geometry as it is in Euclidean geometry, because angle is an example of a concept not invariant under projective transformations, as is seen clearly in perspective drawing. One source for projective geometry was indeed the theory of perspective. Another difference from elementary geometry is the way in which parallel lines can be said to meet in a point at infinity, once the concept is translated into projective geometry's terms. Again this notion has an intuitive basis, such as railway tracks meeting at the horizon in a perspective drawing. See projective plane for the basics of projective geometry in two dimensions.While the ideas were available earlier, projective geometry was mainly a development of the nineteenth century. A huge body of research made it the most representative field of geometry of that time. This was the theory of complex projective space, since the coordinates used (homogeneous coordinates) were complex numbers. Several major strands of more abstract mathematics (including invariant theory, the Italian school of algebraic geometry, and Felix Klein's Erlangen programme leading to the study of the classical groups) built on projective geometry. It was also a subject with a large number of practitioners for its own sake, under the banner of synthetic geometry. Another field that emerged from axiomatic studies of projective geometry is finite geometry.The field of projective geometry is itself now divided into many research subfields, two examples of which are projective algebraic geometry (the study of projective varieties) and projective differential geometry (the study of differential invariants of the projective transformations).".
- Projective_geometry wikiPageExternalLink summary?doi=10.1.1.17.1329.
- Projective_geometry wikiPageExternalLink isprs96.html.
- Projective_geometry wikiPageExternalLink burali-forti_-_diff._geom._following_grassmann.pdf.
- Projective_geometry wikiPageExternalLink burali-forti_-_grassman_and_proj._geom..pdf.
- Projective_geometry wikiPageExternalLink kummer_-_rectilinear_ray_systems.pdf.
- Projective_geometry wikiPageExternalLink pasch_-_focal_and_singularity_surfaces.pdf.
- Projective_geometry wikiPageExternalLink projective.pdf.
- Projective_geometry wikiPageExternalLink ?p=262.
- Projective_geometry wikiPageExternalLink projective_geometry.html.
- Projective_geometry wikiPageID "243849".
- Projective_geometry wikiPageRevisionID "603890397".
- Projective_geometry hasPhotoCollection Projective_geometry.
- Projective_geometry subject Category:Projective_geometry.
- Projective_geometry comment "In mathematics, projective geometry is the study of geometric properties that are invariant under projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts.".
- Projective_geometry label "Geometria proiettiva".
- Projective_geometry label "Geometria projetiva".
- Projective_geometry label "Geometria rzutowa".
- Projective_geometry label "Geometría proyectiva (Matemáticas)".
- Projective_geometry label "Géométrie projective".
- Projective_geometry label "Projectieve meetkunde".
- Projective_geometry label "Projective geometry".
- Projective_geometry label "Projektive Geometrie".
- Projective_geometry label "Проективная геометрия".
- Projective_geometry label "هندسة إسقاطية".
- Projective_geometry label "射影几何".
- Projective_geometry label "射影幾何学".
- Projective_geometry sameAs Projektivní_geometrie.
- Projective_geometry sameAs Projektive_Geometrie.
- Projective_geometry sameAs Geometría_proyectiva_(Matemáticas).
- Projective_geometry sameAs Géométrie_projective.
- Projective_geometry sameAs Geometri_projektif.
- Projective_geometry sameAs Geometria_proiettiva.
- Projective_geometry sameAs 射影幾何学.
- Projective_geometry sameAs 사영기하학.
- Projective_geometry sameAs Projectieve_meetkunde.
- Projective_geometry sameAs Geometria_rzutowa.
- Projective_geometry sameAs Geometria_projetiva.
- Projective_geometry sameAs m.01kbq3.
- Projective_geometry sameAs Q177409.
- Projective_geometry sameAs Q177409.
- Projective_geometry wasDerivedFrom Projective_geometry?oldid=603890397.
- Projective_geometry isPrimaryTopicOf Projective_geometry.