Translation plane

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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, $\scriptstyle p$ 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.^[2]

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).

[edit] Relationship to spreads

Translation planes are related to spreads in finite projective spaces by the André/Bruck-Bose construction.^[3] A spread of $\scriptstyle PG(3, q)$ is a set of q² + 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 q² 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 q². Let $\scriptstyle \pi(S)$ be the projective completion of $\scriptstyle A(S)$ .^[4]^[5]

[edit] References

^ Projective Planes On projective planes
^ Geometry Translation Plane Retrieved on June 13, 2007
^ Finite Geometry André/Bruck-Bose Retrieved on May 24, 2007
^ André, Johannes (1954). Über nicht-Dessarguessche Ebenen mit transitiver Translationsgruppe, pp. 156-186.
^ Bruck, R. H.; R. C. Bose (1964). The Construction of Translation Planes from Projective Spaces, pp. 85-102.