Universality theorems for configuration spaces of planar linkages

Planar Linkages and Algebraic Sets

A linkage is a finite graph with lengths assigned to each edge. A planar realization is a map to the plane which preserves edge lengths. It can be thought of as a mechanical device formed from stiff rods and rotating joints. We look at the configuration space of all planar realizations of a linkage (following work of Kapovich-Millson). We also look at configuration spaces of cabled linkages, where some edges are flexible cables. These configuration spaces are classified up to analytic isomorphism.

Semiconfiguration Spaces of Planar Linkages

This paper characterizes which subsets of C^n can be the set of positions of n points on a linkage in the complex plane C. For example, assuming compactness they are just compact semialgebraic sets. Noncompact configuration spaces are semialgebraics sets invariant under the Euclidean group, with compact quotient.

五边形连杆机构的双心构型

Bicentric configurations of pentagonal linkages

本文研究平面多边形连杆机构存在双心构型的边长条件，给出切接与弦接多边形的相关结果，并完整刻画了具有双心构型的五边形连杆机构。

平面连杆配置曲线的亏格通常是奇数

The genus of configuration curves of planar linkages is generically odd

本文证明平面连杆配置曲线的亏格在一般情况下为奇数，基于热带几何方法。

Planar Linkages Following a Prescribed Motion

Designing mechanical devices, called linkages, that draw a given plane curve has been a topic that interested engineers and mathematicians for hundreds of years, and recently also computer scientists. Already in 1876, Kempe proposed a procedure for solving the problem in full generality, but his constructions tend to be extremely complicated. We provide a novel algorithm that produces much simpler linkages, but works only for parametric curves. Our approach is to transform the problem into a factorization task over some noncommutative algebra. We show how to compute such a factorization, and h…

Kempe's Universality Theorem for Rational Space Curves

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.