Homology, Homotopy and Applications

Volume 6 (2004)

Number 1

The geometry of configuration spaces for closed chains in two and three dimensions

Pages: 237 – 267

DOI: https://dx.doi.org/10.4310/HHA.2004.v6.n1.a14

Authors

R. James Milgram (Department of Mathematics, Stanford University, Stanford, Calif., U.S.A.)

J. C. Trinkle (Department of Computer Science, Rensselaer Polytechnic Institute, Troy, New York, U.S.A.)

Abstract

In this note we analyze the topology of the spaces of configurations in the euclidian space $\mathbb{R}^n$ of all linearly immersed polygonal circles with either fixed lengths for the sides or one side allowed to vary. Specifically, this means that the allowed maps of a $k$-gon $\langle l_1, l_2, \dots, l_k\rangle$ where the $l_i$ are the lengths of the successive sides, are specified by an ordered $k$-tuple of points in $\mathbb{R}^n$, $P_1,~P_2, \dots, P_k$ with $d(P_i, P_{i+1}) = l_i$, $1 \le i \le k-1$ and $d(P_k, P_1) = l_k$. The most useful cases are when $n = 2$ or $3$, but there is no added complexity in doing the general case. In all dimensions, we show that the configuration spaces are manifolds built out of unions of specific products $(S^{n-1})^H\times I^{(n-1)(k-2 -H)}$, over (specific) common sub-manifolds of the same form or the boundaries of such manifolds. Once the topology is specified, it is indicated how to apply these results to motion planning problems in $\mathbb{R}^2$.

Keywords

configuration spaces, linkages

2010 Mathematics Subject Classification

55R80

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Published 1 January 2004