Cubic pencils of lines and bivariate interpolation

Cubic pencils of lines are classified up to projectivities. Explicit formulae for the addition of lines on the set of nonsingular lines of the pencils are given. These formulae can be used for constructing planar generalized principal lattices, which are sets of points giving rise to simple Lagrange formulae in bivariate interpolation. Special attention is paid to the irreducible nonsingular case, where elliptic functions are used in order to express the addition in a natural form.