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- Incidence_(geometry) abstract "In geometry, an incidence relation is a binary relation between different types of objects that captures the idea being expressed when phrases such as \"a point lies on a line\" or \"a line is contained in a plane\" are used. The most basic incidence relation is that between a point, P, and a line, l, sometimes denoted P I l. If P I l the pair (P, l) is called a flag. There are many expressions used in common language to describe incidence (for example, a line passes through a point, a point lies in a plane, etc.) but the term \"incidence\" is preferred because it does not have the additional connotations that these other terms have, and it can be used in a symmetric manner, reflecting this property of the relation. Statements such as \"line l1 intersects line l2\" are also statements about incidence relations, but in this case, it is because this is a shorthand way of saying that \"there exists a point P that is incident with both line l1 and line l2\". When one type of object can be thought of as a set of the other type of object (viz., a plane is a set of points) then an incidence relation may be viewed as containment.Statements such as \"any two lines in a plane meet\" are called incidence propositions. This particular statement is true in a projective plane, though not true in the Euclidean plane where lines may be parallel. Historically, projective geometry was developed in order to make the propositions of incidence true without exceptions, such as those caused by the existence of parallels. From the point of view of synthetic geometry, projective geometry should be developed using such propositions as axioms. This is most significant for projective planes due to the universal validity of Desargues' theorem in higher dimensions. In contrast, the analytic approach is to define projective space based on linear algebra and utilizing homogeneous co-ordinates. The propositions of incidence are derived from the following basic result on vector spaces: given subspaces U and W of a (finite dimensional) vector space V, the dimension of their intersection is dim U + dim W − dim (U + W). Bearing in mind that the geometric dimension of the projective space P(V) associated to V is dim V − 1 and that the geometric dimension of any subspace is positive, the basic proposition of incidence in this setting can take the form: linear subspaces L and M of projective space P meet provided dim L + dim M ≥ dim P.The following sections are limited to projective planes defined over fields, often denoted by PG(2, F), where F is a field, or P2F. However these computations can be naturally extended to higher dimensional projective spaces and the field may be replaced by a division ring (or skewfield) provided that one pays attention to the fact that multiplication is not commutative in that case.".
- Incidence_(geometry) wikiPageID "387224".
- Incidence_(geometry) wikiPageLength "12348".
- Incidence_(geometry) wikiPageOutDegree "39".
- Incidence_(geometry) wikiPageRevisionID "697233632".
- Incidence_(geometry) wikiPageWikiLink Axiom.
- Incidence_(geometry) wikiPageWikiLink Binary_relation.
- Incidence_(geometry) wikiPageWikiLink Category:Articles_containing_proofs.
- Incidence_(geometry) wikiPageWikiLink Category:Projective_geometry.
- Incidence_(geometry) wikiPageWikiLink Cevas_theorem.
- Incidence_(geometry) wikiPageWikiLink Commutative_property.
- Incidence_(geometry) wikiPageWikiLink Concurrent_lines.
- Incidence_(geometry) wikiPageWikiLink Concyclic_points.
- Incidence_(geometry) wikiPageWikiLink Desarguess_theorem.
- Incidence_(geometry) wikiPageWikiLink Determinant.
- Incidence_(geometry) wikiPageWikiLink Division_ring.
- Incidence_(geometry) wikiPageWikiLink Dot_product.
- Incidence_(geometry) wikiPageWikiLink Field_(mathematics).
- Incidence_(geometry) wikiPageWikiLink Geometry.
- Incidence_(geometry) wikiPageWikiLink Hilberts_axioms.
- Incidence_(geometry) wikiPageWikiLink Homogeneous_coordinates.
- Incidence_(geometry) wikiPageWikiLink Incidence_algebra.
- Incidence_(geometry) wikiPageWikiLink Incidence_geometry.
- Incidence_(geometry) wikiPageWikiLink Incidence_matrix.
- Incidence_(geometry) wikiPageWikiLink Incidence_structure.
- Incidence_(geometry) wikiPageWikiLink Levi_graph.
- Incidence_(geometry) wikiPageWikiLink Linear_algebra.
- Incidence_(geometry) wikiPageWikiLink Linear_subspace.
- Incidence_(geometry) wikiPageWikiLink Menelaus_theorem.
- Incidence_(geometry) wikiPageWikiLink Parallel_(geometry).
- Incidence_(geometry) wikiPageWikiLink Prentice_Hall.
- Incidence_(geometry) wikiPageWikiLink Projective_geometry.
- Incidence_(geometry) wikiPageWikiLink Projective_plane.
- Incidence_(geometry) wikiPageWikiLink Projective_range.
- Incidence_(geometry) wikiPageWikiLink Projective_space.
- Incidence_(geometry) wikiPageWikiLink Subset.
- Incidence_(geometry) wikiPageWikiLink Symmetric_relation.
- Incidence_(geometry) wikiPageWikiLink Synthetic_geometry.
- Incidence_(geometry) wikiPageWikiLink Triple_product.
- Incidence_(geometry) wikiPageWikiLink Two-dimensional_space.
- Incidence_(geometry) wikiPageWikiLink Vector_space.
- Incidence_(geometry) wikiPageWikiLinkText "Incidence (geometry)".
- Incidence_(geometry) wikiPageWikiLinkText "Incidence (geometry)#Collinearity".
- Incidence_(geometry) wikiPageWikiLinkText "Incidence Geometry".
- Incidence_(geometry) wikiPageWikiLinkText "Incidence".
- Incidence_(geometry) wikiPageWikiLinkText "collinear".
- Incidence_(geometry) wikiPageWikiLinkText "incidence geometry".
- Incidence_(geometry) wikiPageWikiLinkText "incidence property".
- Incidence_(geometry) wikiPageWikiLinkText "incidence relation".
- Incidence_(geometry) wikiPageWikiLinkText "incidence".
- Incidence_(geometry) wikiPageWikiLinkText "incidences".
- Incidence_(geometry) wikiPageWikiLinkText "incident".
- Incidence_(geometry) wikiPageUsesTemplate Template:!.
- Incidence_(geometry) wikiPageUsesTemplate Template:Main.
- Incidence_(geometry) wikiPageUsesTemplate Template:Math.
- Incidence_(geometry) wikiPageUsesTemplate Template:Reflist.
- Incidence_(geometry) subject Category:Articles_containing_proofs.
- Incidence_(geometry) subject Category:Projective_geometry.
- Incidence_(geometry) hypernym Relation.
- Incidence_(geometry) type Agent.
- Incidence_(geometry) type Proof.
- Incidence_(geometry) comment "In geometry, an incidence relation is a binary relation between different types of objects that captures the idea being expressed when phrases such as \"a point lies on a line\" or \"a line is contained in a plane\" are used. The most basic incidence relation is that between a point, P, and a line, l, sometimes denoted P I l. If P I l the pair (P, l) is called a flag.".
- Incidence_(geometry) label "Incidence (geometry)".
- Incidence_(geometry) sameAs Q1671857.
- Incidence_(geometry) sameAs Inzidenz_(Geometrie).
- Incidence_(geometry) sameAs Incidenza_(geometria).
- Incidence_(geometry) sameAs Incydencja.
- Incidence_(geometry) sameAs m.022c77.
- Incidence_(geometry) sameAs Incidens_(geometri).
- Incidence_(geometry) sameAs Q1671857.
- Incidence_(geometry) wasDerivedFrom Incidence_(geometry)?oldid=697233632.
- Incidence_(geometry) isPrimaryTopicOf Incidence_(geometry).