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In [[mathematics]], the '''real projective line''' is a [[topological space]] that is [[homeomorphic]] to a [[circle]]. In [[differential geometry]], it is a [[manifold]] that requires more than one [[coordinate chart]] in an [[atlas (topology)|atlas]] to cover it. The real projective line arises in [[algebraic geometry]] as a one-dimensional subspace of the [[real projective plane]] or of the [[complex projective line]]. The [[automorphism]]s of the real projective line form a [[projective linear group]] known as PGL(2,R). |
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#REDIRECT [[Projectively extended real line]] |
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==Definition== |
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{{R from move}} |
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The concepts of [[homogeneous coordinates]], or [[equivalence relation]]s and [[equivalence class]]es, are necessary to define the real projective line. The starting point is the direct product R x R = R<sup>2</sup>, with the point (0,0) excluded. Define the [[binary relation]] (''a,b'') ~ (''c,d'') to hold when ''ad'' = ''bc''. It may be confirmed that this relation is a [[reflexive relation]] and a [[symmetric relation]]. The proof that it is a [[transitive relation]] requires attention to the case ''a'' = 0, but is otherwise an elementary deduction. These three properties imply that it is an equivalence relation; consequently partitioning its space into equivalence classes R(''a,b''). The set of all equivalence classes is P<sup>1</sup>R, the real projective line, with typical point R(''a.b'').<ref>The argument used to construct P<sup>1</sup>R can also be used with any [[field (mathematics)|field]] K to construct P<sup>1</sup>K .</ref> |
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==Charts== |
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Two coordinate charts may be used to parametrize the points of P<sup>1</sup>R: |
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* Chart #1: <math>b\ne 0, \quad R(a,b) \mapsto \frac {a}{b} ,</math> |
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* Chart #2: <math>a\ne 0, \quad R(a,b) \mapsto \frac {b}{a} ,</math>. |
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Note that the equivalence relation provides that all representatives of an equivalence class are sent to the same coordinate in R by a chart. Note also, that either of ''a'' or ''b'' may be zero, but not both, so both charts are needed to cover P<sup>1</sup>R. Furthermore, the mapping from one chart to the other is the [[multiplicative inverse]], a [[differentiable function]]. |
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==Structure== |
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The real projective line is a [[complete space|complete]] [[projective range]] that is found in the real projective plane and in the complex projective line. Its structure is thus inherited from these superstructures.<ref>If a real projective line happens to appear in a [[non-Desarguesian plane]] the harmonic structure cannot be presumed</ref> Primary among these structures is the relation of [[projective harmonic conjugates]] among the points of the projective range. |
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==Automorphisms== |
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The mappings of P<sup>1</sup>R are called [[homography|homographies]] or projectivities. Traditionally these were constructed as [[central projection]]s or [[parallel projection]]s and their compositions. However, with the rise of algebraic geometry and the use of homogeneous coordinates, the automorphisms are expressed with the facility of [[2 × 2 real matrices]], provided that proportional matrices are identified, producing the projective linear group PGL(2,R). This group lies between the [[modular group]] and the [[Möbius group]]. |
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==Notes== |
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{{Reflist}} |
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==References== |
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* Juan Carlos Alvarez (2000) [http://www.math.poly.edu/courses/projective_geometry/chapter_two/chapter_two.html The Real Projective Line], course content from [[New York University]]. |
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* Santiago Cañez (2014) [http://math.northwestern.edu/~scanez/courses/math340/winter14/handouts/projective.pdf Notes on Projective Geometry] from [[Northwestern University]]. |
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[[Category: Projective geometry]] |
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[[Category:Manifolds]] |
Revision as of 20:52, 21 June 2015
In mathematics, the real projective line is a topological space that is homeomorphic to a circle. In differential geometry, it is a manifold that requires more than one coordinate chart in an atlas to cover it. The real projective line arises in algebraic geometry as a one-dimensional subspace of the real projective plane or of the complex projective line. The automorphisms of the real projective line form a projective linear group known as PGL(2,R).
Definition
The concepts of homogeneous coordinates, or equivalence relations and equivalence classes, are necessary to define the real projective line. The starting point is the direct product R x R = R2, with the point (0,0) excluded. Define the binary relation (a,b) ~ (c,d) to hold when ad = bc. It may be confirmed that this relation is a reflexive relation and a symmetric relation. The proof that it is a transitive relation requires attention to the case a = 0, but is otherwise an elementary deduction. These three properties imply that it is an equivalence relation; consequently partitioning its space into equivalence classes R(a,b). The set of all equivalence classes is P1R, the real projective line, with typical point R(a.b).[1]
Charts
Two coordinate charts may be used to parametrize the points of P1R:
- Chart #1:
- Chart #2: .
Note that the equivalence relation provides that all representatives of an equivalence class are sent to the same coordinate in R by a chart. Note also, that either of a or b may be zero, but not both, so both charts are needed to cover P1R. Furthermore, the mapping from one chart to the other is the multiplicative inverse, a differentiable function.
Structure
The real projective line is a complete projective range that is found in the real projective plane and in the complex projective line. Its structure is thus inherited from these superstructures.[2] Primary among these structures is the relation of projective harmonic conjugates among the points of the projective range.
Automorphisms
The mappings of P1R are called homographies or projectivities. Traditionally these were constructed as central projections or parallel projections and their compositions. However, with the rise of algebraic geometry and the use of homogeneous coordinates, the automorphisms are expressed with the facility of 2 × 2 real matrices, provided that proportional matrices are identified, producing the projective linear group PGL(2,R). This group lies between the modular group and the Möbius group.
Notes
- ^ The argument used to construct P1R can also be used with any field K to construct P1K .
- ^ If a real projective line happens to appear in a non-Desarguesian plane the harmonic structure cannot be presumed
References
- Juan Carlos Alvarez (2000) The Real Projective Line, course content from New York University.
- Santiago Cañez (2014) Notes on Projective Geometry from Northwestern University.