  This paper studies the set of equivalent realizations of isostatic frameworks
in two dimensions, and algorithms for finding all such realizations. We show
that an isostatic framework has an even number of equivalent realizations that
preserve edge lengths and connectivity. We enumerate the complete set of
equivalent realizations for a toy framework with pinned boundary in two
dimensions and study the impact of boundary length on the emergence of these
realizations. To ameliorate the computational complexity of finding a solution
to a large multivariate quadratic system corresponding to the constraints;
alternative methods - based on constraint reduction and distance-based covering
map or Cayley parameterization of the search space - are presented. The
application of these methods is studied on atomic clusters, a model
two-dimensional glasses, and jamming.
