Automated generation of Kempe linkages for algebraic curves and surfaces

The well-known Peaucellier linkage converts a circular motion into a linear motion. A more shocking result by Kempe [Proc. LMS (1876) 213] shows that any plane algebraic curve can be delineated by a linkage. In this paper, we improve Kempeʹs result to give a precise algorithm for generating the Kempe linkage. We prove that for an algebraic plane curve of degree n, the corresponding Kempe linkage uses at most O(n4) links. We extend Kempeʹs result to the 3D case: any algebraic surface can be delineated by a 3D linkage. We implement a program which can construct the Kempe linkage and animate the drawing process for a given plane curve.