Kempe's Universality Theorem for Rational Space Curves

TL;DR

This paper proves a universal theorem for drawing any bounded rational space curve using linkages with revolute joints, based on motion polynomial factorization and minimal degree construction.

Contribution

It establishes a universal linkage construction for rational space curves using motion polynomial factorization and explicit linkage design.

Findings

Proves that any bounded rational space curve can be drawn with a specific linkage.

Provides an explicit construction method for the linkage.

Introduces a minimal degree motion polynomial for given orbits.

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.