A characterisation of the generic rigidity of 2-dimensional point–line frameworks

A 2-dimensional point–line framework is a collection of points and lines in the plane which are linked by pairwise constraints that fix some angles between pairs of lines and also some point–line and point–point distances. It is rigid if every continuous motion of the points and lines which preserves the constraints results in a point–line framework which can be obtained from the initial framework by a translation or a rotation. We characterise when a generic point–line framework is rigid. Our characterisation gives rise to a polynomial algorithm for solving this decision problem.