Kempe’s Universality Theorem for Rational Space Curves

Abstract

We prove that every bounded rational space curve of degree d and circularity c can be drawn by a linkage with 9/2 d - 6c + 1 revolute joints. Our proof is based on two ingredients. The first one is the factorization theory of motion polynomials. The second one is the construction of a motion polynomial of minimum degree with given orbit. Our proof also gives the explicity construction of the linkage.