線形並進ばねネットワークの平衡点
Equilibria for Networks of Linear Translational Springs
非線形代数の手法を用いて、2次元・3次元における最大5ノードの剛体線形並進ばねネットワークの平衡点をすべて解き、解の総数に関する次数の上限を与えた。
著者: Luke Oeding, Ethan Clayton, Jackson Elsea, Nicholas Wang, Adam Rutkowski
分類: math.AG, cs.RO
原文アブストラクト
We use tools from nonlinear algebra to study the equilibria of small linear translational spring networks. Specifically we use the techniques of homotopy continuation, monodromy, and parameter homotopy (a.k.a. cheater homotopy) to solve all rigid linear translational spring networks up to $5$ nodes in both $2$ and $3$ dimensions. We describe a method of implementing parameter homotopy that arises naturally from the physical structure of the system. We give precise total degree bounds on the maximum number of solutions for general planar spring networks. We discuss further efficiency gains obtained from polyhedral homotopy methods. We compare the computation efficiency of these techniques against a baseline of Newton's method.