Elementary proofs of Kempe universality

TL;DR

This paper provides elementary proofs demonstrating that algebraic and continuous curves can be traced by finite or infinite linkages, respectively, with bounded joint valencies, and offers corrections to Kempe's original theorems.

Contribution

It introduces simplified proofs for Kempe universality and extends the results to continuous curves with bounded joint valencies, correcting historical inaccuracies.

Findings

Finite pinned linkages can trace algebraic curves.

Infinite linkages can trace continuous curves with bounded joint valencies.

A correction to Kempe's original argument is provided.

Abstract

An elementary proof is given to show that a parametrised algebraic curve in the plane may be traced out, in the sense of A. B. Kempe, by a finite pinned linkage. Additionally it is shown that any parametrised continuous curve \gamma: [0,1] to R^2 may be traced out by an infinite linkage where the valencies of the joints is uniformly bounded. We also discuss related Kempe universality theorems and give a novel correction of Kempe's original argument.