The Geometry of Configuration Spaces for Closed Chains in Two and Three Dimensions
نویسندگان
چکیده
In this note we analyze the topology of the moduli spaces of configurations in the euclidian space R 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 〈l1, l2, . . . , lk〉 where the li are the lengths of the successive sides, are specified by an ordered k-tuple of points in R, P1, P2, . . . , Pk with d(Pi, Pi+1) = li, 1 ≤ i ≤ k − 1 and d(Pk, P1) = lk. The must 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 these configuration spaces are manifolds built out of unions of specific products (S) × I , 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 R.
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