---
abstract: 'We prove that every bounded rational space curve of degree $d$ and circularity $c$ can be drawn by a linkage with $ \frac{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 explicit construction of the linkage.'
address:
- 'Joanneum Research, Institute for Robotics and Mechatronics, Lakeside B08a, 9020 Klagenfurt, Austria, Phone +43 316 876 2016'
- 'Research Institute for Symbolic Computation, Johannes Kepler University Linz, Schloss Hagenberg, 4232 Hagenberg, Austria'
- 'Unit Geometry and CAD, University of Innsbruck, Technikerstr. 13, 6020 Innsbruck, Austria'
author:
- Zijia Li
- Josef Schicho
- 'Hans-Peter Schröcker'
title: |
    Kempe’s Universality Theorem\
    for Rational Space Curves
---

Introduction {#sec:introduction}
============

Kempe’s Universality Theorem [@kempe76] is one of the great theorems of theoretical mechanism science (“beautiful” [@blaschke56; @demaine07], “surprising” [@abbott08; @demaine07], “incredible theoretical significance” [@saxena11], “shocking” [@gao01]). It states that any bounded portion of a planar algebraic curve can be traced out by one joint of a planar linkage with revolute joints. Discovered only shortly after the invention of the 