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

Abstract

In this note we analyze the topology of the spaces of configurations in the euclidian space $\bbr^n$ 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 $\bbr^n$, $P_1,~P_2, \dots, P_k$ 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 $\bbr^2$.