Continuous curves from infinite Kempe linkages

Abstract

In 1876 Kempe showed that any algebraic curve in the plane may be realised as the locus of one of the joints of a finite bar-joint linkage. An often cited illustration of this is that there is a linkage that can write a person's signature to any particular accuracy. An infinite analogue is established showing that any continuous curve in the plane is the curve of motion of a joint of an infinite bar-joint linkage. This is curious, as continuous curves can be space filling. Moreover, there is a single infinite linkage that simultaneously traces everybody's signature with no error whatsoever.