Geometric characterization and generalized principal lattices

The sets of nodes in the plane for which its nth degree Lagrange polynomials can be factored as a product of first degree polynomials satisfy a geometric characterization: for each node there exists a set of ≤n lines containing the other nodes. Generalized principal lattices are sets of nodes defined by three families of lines. A generalized principal lattice satisfies the geometric characterization and there exist exactly three lines in the plane containing more nodes than the degree. In this paper, we show a converse, valid for degrees n≤7: if a set of nodes satisfy the geometric characterization and there exist exactly three lines containing n+1 nodes, then it is a generalized principal lattice.