Synthesis of a linkage to draw a plane algebraic curve

- Presentation of a new method for the design of linkages derived from Kempe's Universality Theorem.
- Demonstration that bevel gear differentials and cable drives can simplify the linkages obtained to draw a plane algebraic curve.
- The method is demonstrated on the design of mechanisms to draw two straight lines and an elliptic cubic curve.
- A comparison of part counts shows the new method reduces the number of parts by half.

This paper presents a synthesis methodology for a planar linkage that draws a given plane algebraic curve. The existence of such a linkage was demonstrated by A. B. Kempe, who used by specialized linkages, known as additor, mutiplicator and translator linkages, to constrain the two joints of a planar two-link serial chain so that its end point traced the desired curve. Recent research has verified Kempe's results, but yield linkages that are exceedingly complex. In this paper, we replace Kempe's specialized linkages with differentials and cable drives that perform the same functions but simplify the linkage. The result is a construction of Kempe's drawing linkage that illustrates the underlying structure of his approach. Examples are provided that illustrate the theory.