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 contin
uous 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.
A 2-dimensional framework is a straight line realisation of a graph in the Euclidean plane. It is radically solvable if the set of vertex coordinates is contained in a radical extension of the field of rationals extended by the squared edge lengths.
We show that the radical solvability of a generic framework depends only on its underlying graph and characterise which planar graphs give rise to radically solvable generic frameworks. We conjecture that our characterisation extends to all graphs.
Given a rigid realisation of a graph $G$ in ${mathbb R}^2$, it is an open problem to determine the maximum number of pairwise non-congruent realisations which have the same edge lengths as the given realisation. This problem can be restated as findin
g the number of solutions of a related system of quadratic equations and in this context it is natural to consider the number of solutions in ${mathbb C}^2$ rather that ${mathbb R}^2$. We show that the number of complex solutions, $c(G)$, is the same for all generic realisations of a rigid graph $G$, characterise the graphs $G$ for which $c(G)=1$, and show that the problem of determining $c(G)$ can be reduced to the case when $G$ is $3$-connected and has no non-trivial $3$-edge-cuts. We consider the effect of the Henneberg moves and the vertex-splitting operation on $c(G)$. We use our results to determine $c(G)$ exactly for two important families of graphs, and show that the graphs in both families have $c(G)$ pairwise equivalent generic real realisations. We also show that every planar isostatic graph on $n$ vertices has at least $2^{n-3}$ pairwise equivalent real realisations.
