Rigid components in fixed-lattice and cone frameworks

We study the fundamental algorithmic rigidity problems for generic frameworks periodic with respect to a fixed lattice or a finite-order rotation in the plane. For fixedlattice frameworks we give an O(n) algorithm for deciding generic rigidity and an O(n) algorithm for computing rigid components. If the order of rotation is part of the input, we give an O(n) algorithm for deciding rigidity; in the case where the rotation’s order is 3, a more specialized algorithm solves all the fundamental algorithmic rigidity problems in O(n) time.