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In mathematics, a translation plane is a particular kind of projective plane, as considered as a combinatorial object.^[1]

In a projective plane, $p$ represents a point, and $L$ represents a line. A central collineation with center $p$ and axis $L$ is a collineation fixing every point on $L$ and every line through $p$ . It is called an "elation" if $p$ is on $L$ , otherwise it is called a "homology". The central collineations with centre $p$ and axis $L$ form a group.^[2]

A projective plane $Π$ is called a translation plane if there exists a line $L$ such that the group of elations with axis $L$ is transitive on the affine plane Π_l (the affine derivative of Π).

Relationship to spreads

Translation planes are related to spreads in finite projective spaces by the André/Bruck-Bose construction.^[3] A spread of $P G (3, q)$ is a set of q² + 1 lines, with no two intersecting. Equivalently, it is a partition of the points of $P G (3, q)$ into lines.

Given a spread $S$ of $P G (3, q)$ , the André/Bruck-Bose construction¹ produces a translation plane $π (S)$ of order q² as follows: Embed $P G (3, q)$ as a hyperplane of $P G (4, q)$ . Define an incidence structure $A (S)$ with "points," the points of $P G (4, q)$ not on $P G (3, q)$ and "lines" the planes of $P G (4, q)$ meeting $P G (3, q)$ in a line of $S$ . Then $A (S)$ is a translation affine plane of order q². Let $π (S)$ be the projective completion of $A (S)$ .^[4]^[5]

References

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External links

↑ Projective Planes On projective planes
↑ Geometry Translation Plane Retrieved on June 13, 2007
↑ Template:Cite web
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↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

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[1] Projective Planes On projective planes

[2] Geometry Translation Plane Retrieved on June 13, 2007

[3] Template:Cite web

[4] 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

[5] 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

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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>
+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>
+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).
+== Relationship to spreads ==
+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>&nbsp;+&nbsp;1 lines, with no two intersecting. Equivalently, it is a partition of the points of <math>\scriptstyle PG(3, q) </math> into lines.
+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
+  | 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
+  | 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>
+==References==
+{{Reflist}}
+==Further reading==
+* Mauro Biliotti, Vikram Jha, Norman L. Johnson (2001) ''Foundations of Translation Planes'', [[Marcel Dekker]] ISBN 0-8247-0609-9 .
+==External links==
+*[http://www.library.tuiasi.ro/ipm/vol13no34/pure.html  Foundations_of_Translation_Planes]
+*[http://www-math.ucdenver.edu/~wcherowi/courses/m6221/pglc3a.html Lecture Notes on Projective Geometry]
+*[http://mellinger.umwblogs.org/publications/ Publications of Keith Mellinger]
+{{DEFAULTSORT:Translation Plane}}
+[[Category:Projective geometry]]

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Revision as of 19:12, 28 October 2013

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Relationship to spreads

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Further reading

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