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| In [[mathematics]], a '''translation plane''' is a particular kind of [[projective plane]], as considered as a combinatorial object.<ref>Projective Planes [http://www.maths.qmul.ac.uk/~pjc/pps/pps2.pdf On projective planes]</ref>
| | Golda is what's written on my birth certification even though it is not the title on my birth certificate. Credit authorising is how he tends to make money. For a whilst I've been in Alaska but I will have to move in a year or two. Doing ballet is some thing she would never give up.<br><br>Review my web site :: [http://www.aseandate.com/index.php?m=member_profile&p=profile&id=13352970 love psychics] |
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| In a projective plane, <math>\scriptstyle p</math> represents a point, and <math>\scriptstyle L</math> represents a line. A central [[collineation]] with center <math>\scriptstyle p</math> and axis <math>\scriptstyle L</math> is a collineation fixing every point on <math>\scriptstyle L</math> and every line through <math>\scriptstyle p</math>. It is called an "elation" if <math>\scriptstyle p</math> is on <math>\scriptstyle L</math>, otherwise it is called a "homology". The central collineations with centre <math>\scriptstyle p</math> and axis <math>\scriptstyle L</math> form a group.<ref>Geometry [http://www.math.uni-kiel.de/geometrie/klein/math/geometry/translation.html Translation Plane] Retrieved on June 13, 2007</ref>
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| A projective plane <math>\scriptstyle \Pi</math> is called a translation plane if there exists a line <math>\scriptstyle L</math> such that the group of elations with axis <math>\scriptstyle L</math> is transitive on the affine plane Π<sub>l</sub> (the [[Affine geometry|affine]] derivative of Π).
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| == Relationship to spreads ==
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| Translation planes are related to spreads in finite projective spaces by the André/Bruck-Bose construction.<ref>{{cite web|url=http://www-ma4.upc.es/~simeon/bblpsympspread.pdf|title=Symplectice Spreads|last=Ball|first=Simeon|coauthors=John Bamberg, Michel Lavrauw, Tim Penttila|date=2003-09-15|publisher=[[Polytechnic University of Catalonia]]|accessdate=2008-10-08}}</ref> A spread of <math>\scriptstyle PG(3, q) </math> is a set of ''q''<sup>2</sup> + 1 lines, with no two intersecting. Equivalently, it is a partition of the points of <math>\scriptstyle PG(3, q) </math> into lines.
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| Given a spread <math>\scriptstyle S</math> of <math>\scriptstyle PG(3, q) </math>, the André/Bruck-Bose construction<sup>1</sup> produces a translation plane <math>\scriptstyle \pi(S)</math> of order ''q''<sup>2</sup> as follows: Embed <math>\scriptstyle PG(3, q) </math> as a hyperplane of <math>\scriptstyle PG(4, q) </math>. Define an incidence structure <math>\scriptstyle A(S)</math> with "points," the points of <math>\scriptstyle PG(4, q) </math> not on <math>\scriptstyle PG(3, q) </math> and "lines" the planes of <math>\scriptstyle PG(4, q) </math> meeting <math>\scriptstyle PG(3, q) </math> in a line of <math>\scriptstyle S</math>. Then <math>\scriptstyle A(S)</math> is a translation affine plane of order ''q''<sup>2</sup>. Let <math>\scriptstyle \pi(S)</math> be the projective completion of <math>\scriptstyle A(S)</math>.<ref>{{cite book
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| | last =André | first =Johannes | authorlink = | coauthors = | title = Über nicht-Dessarguessche Ebenen mit transitiver Translationsgruppe | publisher = | year =1954 | location = | pages =156–186 | url = | doi = | id = }}</ref><ref>{{cite book
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| | last =Bruck | first = R. H. | authorlink = Richard Bruck| coauthors = R. C. Bose | title = The Construction of Translation Planes from Projective Spaces | publisher = | year =1964 | location = | pages = 85–102 | url = | doi = | id = }}</ref>
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| ==References==
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| {{Reflist}}
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| ==Further reading==
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| * Mauro Biliotti, Vikram Jha, Norman L. Johnson (2001) ''Foundations of Translation Planes'', [[Marcel Dekker]] ISBN 0-8247-0609-9 .
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| ==External links==
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| *[http://www.library.tuiasi.ro/ipm/vol13no34/pure.html Foundations_of_Translation_Planes]
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| *[http://www-math.ucdenver.edu/~wcherowi/courses/m6221/pglc3a.html Lecture Notes on Projective Geometry]
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| *[http://mellinger.umwblogs.org/publications/ Publications of Keith Mellinger]
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| {{DEFAULTSORT:Translation Plane}}
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| [[Category:Projective geometry]]
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Golda is what's written on my birth certification even though it is not the title on my birth certificate. Credit authorising is how he tends to make money. For a whilst I've been in Alaska but I will have to move in a year or two. Doing ballet is some thing she would never give up.
Review my web site :: love psychics