The geometry of configuration spaces for closed chains in two and three dimensions
R. James Milgram, Jeff Trinkle · Homology Homotopy and Applications · 2004
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 in R n , P 1 , P 2 , . . ., P k with d(P i , P i+1 ) = l i , 1 i 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 × 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 .