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

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.

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. 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 $\scriptstyle A(S)$.

References

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Further reading

• Mauro Biliotti, Vikram Jha, Norman L. Johnson (2001) Foundations of Translation Planes, Marcel Dekker ISBN 0-8247-0609-9 .

External links

• Foundations_of_Translation_Planes
• Lecture Notes on Projective Geometry
• Publications of Keith Mellinger