Planar Linkages Following a Prescribed Motion

TL;DR

This paper introduces a new algorithm for designing simpler planar linkages that follow prescribed parametric curves by transforming the problem into a noncommutative algebra factorization task.

Contribution

The paper presents a novel algebraic approach to linkage design, simplifying the construction process for parametric curves compared to classical methods.

Findings

The algorithm produces simpler linkages than previous methods.

It effectively computes factorizations in noncommutative algebra.

The approach is limited to parametric curves.

Abstract

Designing mechanical devices, called linkages, that draw a given plane curve has been a topic that interested engineers and mathematicians for hundreds of years, and recently also computer scientists. Already in 1876, Kempe proposed a procedure for solving the problem in full generality, but his constructions tend to be extremely complicated. We provide a novel algorithm that produces much simpler linkages, but works only for parametric curves. Our approach is to transform the problem into a factorization task over some noncommutative algebra. We show how to compute such a factorization, and how to use it to construct a linkage tracing a given curve.