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.
Jackson, B., & Owen, J. C. (2016). A characterisation of the generic rigidity of 2-dimensional point-line frameworks. Journal of Combinatorial Theory. Series B, 119, 96–121. https://doi.org/10.1016/j.jctb.2015.12.007