doi:10.1112/blms/bdp087 Continuous curves from infinite Kempe linkages

Owen and Stephen Power 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. 1