Translation plane

Remarks

Example: let $F$ be a field and let $\Pi$ be the projective plane over $F$, with $\ell$ the line at infinity. For $(a, b) \in F^2$, the translation $(x, y) \mapsto (x + a, y + b)$ of $F^2$ maps lines to parallel lines, so it extends to a collineation of $\Pi$ that fixes every point at infinity. If $(a, b) \ne (0, 0)$, it maps each line of direction $(a, b)$ to itself. So it fixes every line through the point at infinity $C$ of that direction, and it is an elation with axis $\ell$. The translation by $Q - P$ maps $P$ to $Q$, so $\Pi$ is a translation plane with respect to $\ell$.

Non-example: the Hughes plane of order $9$ is a projective plane that is not a translation plane with respect to any of its lines.

When $\Pi$ is a translation plane with respect to $\ell$, the elations with axis $\ell$ form an abelian group that acts regularly on the points off $\ell$. The translation planes are exactly the planes coordinatized by quasifields.