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In this note we analyze the topology of the spaces of configurations in the euclidian space 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 l 1 , l 2 , . . . , l k where the l i are the lengths of the successive sides, are specified by an ordered ktuple of points inThe 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 × 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 R 2 .
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