Rigid components in fixed-lattice and cone frameworks

TL;DR

This paper develops efficient algorithms to determine the rigidity and compute rigid components of periodic frameworks in the plane, considering fixed lattices and finite-order rotations, with special cases optimized for certain symmetries.

Contribution

It introduces new polynomial-time algorithms for rigidity analysis of periodic frameworks with fixed lattices and rotations, including specialized solutions for order 3 rotations.

Findings

O(n^2) algorithm for fixed-lattice rigidity

O(n^3) algorithm for rigid component computation

O(n^4) algorithm for general rotation order rigidity

Abstract

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.