Coupler curves of moving graphs and counting realizations of rigid graphs

TL;DR

This paper studies the geometric properties of calligraphs, a type of moving graph, and establishes a method to count their realizations, leading to improved algorithms and insights into coupler curve invariants.

Contribution

It introduces a vector invariant for calligraphs, relates the number of realizations of certain graphs to an inner product of these vectors, and improves realization counting algorithms.

Findings

The number of realizations equals an inner product of calligraph vectors.

A graph with finitely many realizations can be decomposed into two calligraphs.

The new algorithm enhances counting of realizations and characterizes coupler curve invariants.

Abstract

A calligraph is a graph that for almost all edge length assignments moves with one degree of freedom in the plane, if we fix an edge and consider the vertices as revolute joints. The trajectory of a distinguished vertex of the calligraph is called its coupler curve. To each calligraph we uniquely assign a vector consisting of three integers. This vector bounds the degrees and geometric genera of irreducible components of the coupler curve. A graph, that up to rotations and translations admits finitely many, but at least two, realizations into the plane for almost all edge length assignments, is a union of two calligraphs. We show that this number of realizations is equal to a certain inner product of the vectors associated to these two calligraphs. As an application we obtain an improved algorithm for counting numbers of realizations, and by counting realizations we characterize invariants of coupler curves.