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Statements

Subject Item

dbr:Projective_range

rdfs:label

Projective range

rdfs:comment

In mathematics, a projective range is a set of points in projective geometry considered in a unified fashion. A projective range may be a projective line or a conic. A projective range is the dual of a pencil of lines on a given point. For instance, a correlation interchanges the points of a projective range with the lines of a pencil. A projectivity is said to act from one range to another, though the two ranges may coincide as sets.

dcterms:subject

dbc:Projective_geometry

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34644484

dbo:wikiPageRevisionID

1115153988

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dbr:Karl_von_Staudt dbc:Projective_geometry dbr:Conic_section dbr:Pencil_(mathematics) dbr:Hyperbolic_angle dbr:Projective_harmonic_conjugate dbr:Correlation_(projective_geometry) dbr:Projectivity dbr:Mathematics dbr:H._S._M._Coxeter dbr:University_of_Toronto_Press dbr:Angle dbr:Projective_duality dbr:Julian_Coolidge dbr:Projective_geometry dbr:Real_projective_line dbr:Hyperbola dbr:Circle

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In mathematics, a projective range is a set of points in projective geometry considered in a unified fashion. A projective range may be a projective line or a conic. A projective range is the dual of a pencil of lines on a given point. For instance, a correlation interchanges the points of a projective range with the lines of a pencil. A projectivity is said to act from one range to another, though the two ranges may coincide as sets. A projective range expresses projective invariance of the relation of projective harmonic conjugates. Indeed, three points on a projective line determine a fourth by this relation. Application of a projectivity to this quadruple results in four points likewise in the harmonic relation. Such a quadruple of points is termed a harmonic range. In 1940 Julian Coolidge described this structure and identified its originator: Two fundamental one-dimensional forms such as point ranges, pencils of lines, or of planes are defined as projective, when their members are in one-to-one correspondence, and a harmonic set of one ... corresponds to a harmonic set of the other. ... If two one-dimensional forms are connected by a train of projections and intersections, harmonic elements will correspond to harmonic elements, and they are projective in the sense of Von Staudt.

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wikipedia-en:Projective_range?oldid=1115153988&ns=0

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2892

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wikipedia-en:Projective_range