Planar Linkages Following a Prescribed Motion
This work addresses a long-standing challenge in mechanical design and computer science by simplifying linkage constructions for engineers and mathematicians, though it is incremental as it applies only to parametric curves.
The authors tackled the problem of designing mechanical linkages that draw a given plane curve by developing a novel algorithm that produces simpler linkages for parametric curves, transforming the problem into a factorization task over a noncommutative algebra and demonstrating its construction.
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.