On arachnoid mechanisms formed by concatenation of arc-length parametrized curves

Abstract

We consider arachnoid mechanisms which are formed by concatenation of arc-length parametrized curves in R³ such that the initial point of each curve is fixed to a vertex of a convex polytope P. We prove that the configuration spaces of such arachnoid mechanisms are described in terms of the polar polytope P^Δ. Using this, we determine the homotopy type of the configuration spaces when P is a tetrahedron, cube or octahedron.