A remark about the directed line-elements in the plane (Q1085442)

Consider sets consisting of elements of the form (p,l) where p ranges over a given planar lattice L and for each p the symbol l denotes a halfline starting at p. The distance between two elements (p,l), (q,m) is the sum of the euclidean distance of p and q and the angle between l and m. The author determines all such systems for which the underlying lattice has given determinant \(\geq \pi^ 2/(3-\sqrt{8})\) and for which the minimum distance between its elements is as large as possible, thus solving a problem of L. Fejes Tóth.