Translation plane

In mathematics, a translation plane is a particular kind of projective plane, as considered as a combinatorial object. [Projective Planes [http://www.maths.qmul.ac.uk/~pjc/pps/pps2.pdf On projective planes] ]

In a projective plane, $scriptstyle\; p$ represents a point, and $scriptstyle\; L$ represents a line. A central collineation with center $scriptstyle\; p$ and axis $scriptstyle\; L$ is a collineation fixing every point on $scriptstyle\; L$ and every line through $scriptstyle\; p$. It is called an "elation" if $scriptstyle\; p$ is on $scriptstyle\; L$, otherwise it is called a "homology". The central collineations with centre $scriptstyle\; p$ and axis $scriptstyle\; L$ form a group. [Geometry [http://www.math.uni-kiel.de/geometrie/klein/math/geometry/translation.html Translation Plane] Retrieved on June 13, 2007]

A projective plane $scriptstyle\; Pi$ is called a translation plane if there exists a line $scriptstyle\; L$ such that the group of elations with axis $scriptstyle\; L$ 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. [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] A spread of $scriptstyle\; PG(3,\; q)$ is a set of "q"² + 1 lines, with no two intersecting. Equivalently, it is a partition of the points of $scriptstyle\; PG(3,\; q)$ into lines.

Given a spread $scriptstyle\; S$ of $scriptstyle\; PG(3,\; q)$, the André/Bruck-Bose construction¹ produces a translation plane $scriptstyle\; pi(S)$ of order "q"² as follows: Embed $scriptstyle\; PG(3,\; q)$ as a hyperplane of $scriptstyle\; PG(4,\; q)$. Define an incidence structure $scriptstyle\; A(S)$ with "points," the points of $scriptstyle\; PG(4,\; q)$ not on $scriptstyle\; PG(3,\; q)$ and "lines" the planes of $scriptstyle\; PG(4,\; q)$ meeting $scriptstyle\; PG(3,\; q)$ in a line of $scriptstyle\; S$. Then $scriptstyle\; A(S)$ is a translation affine plane of order "q"². Let $scriptstyle\; pi(S)$ be the projective completion of $scriptstyle\; A(S)$. [cite book
last =André | first =Johannes | authorlink = | coauthors = | title = Über nicht-Dessarguessche Ebenen mit transitiver Translationsgruppe | publisher = | year =1954 | location = | pages =pp. 156-186 | url = | doi = | id = ] [cite book
last =Bruck | first = R. H. | authorlink = | coauthors = R. C. Bose | title = The Construction of Translation Planes from Projective Spaces | publisher = | year =1964 | location = | pages = pp. 85-102 | url = | doi = | id = ]

References

External links

* [http://www.library.tuiasi.ro/ipm/vol13no34/pure.html Foundations_of_Translation_Planes]
* [http://www-math.cudenver.edu/~wcherowi/courses/m6221/pglc3a.html Lecture Notes on Projective Geometry]
* [http://people.umw.edu/~kmelling/pub.html Publications of Keith Mellinger]