# Rigid components in fixed-lattice and cone frameworks

Authors: | Matthew Berardi, Brent Heeringa, Justin Malestein, and Louis Theran |

Proc. of: | Canadian Conference on Computational Geometry CCCG’11, 2011. |

Full text: | arXiv • URL |

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 fixed-lattice frameworks we give an \(O(n^2)\) algorithm for deciding generic rigidity and an \(O(n^3)\) algorithm for computing rigid components. If the order of rotation is part of the input, we give an \(O(n^4)\) 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 \(O(n^2)\) time.