Spaces of Polygons in the Plane and Morse Theory

Abstract

All use subject to http://about.jstor.org/terms uation in which so many "textbook" techniques can be applied. The only pieces of background material that do lie outside the standard undergraduate curriculum are the principles from Morse theory alluded to earlier, so these are introduced and explained

Series parallel linkages

“…The basic idea is to use Morse theory to analyse the fibrewise structure of θ. We remark that the differential properties of the map θ are well understood (see [SV05]). It is differentiable at all points not in θ −1 (0).…”

Connecting Cycles for Concentric Circles

“…Using this formula one can compute the Hessian matrix of L for arbitrary n and verify that it has a tridiagonal form and its determinant gives rise to rational expressions in the radii (for details see n=3 and n=4 below). This determinant can be explicitly evaluated at parades, which gives an expression like the hessian formula given in [5] and yields our second main result.…”

“…Perimeter L(P) of a Γ-circuit P defines a smooth function on P(Γ). A classical problem of combinatorial geometry is to find a point of global minimum of L on p(Γ)which is called a minimal connecting cycle [5]. We extend this problem by considering all critical points of L referred to as stationary connecting cycles for Γ or stationary Γ-circuits.…”

On the moduli space of equilateral plane pentagons

“…Moreover, the moduli space depends on the choice of the lengths d k and in the smooth case is diffeomorphic to one of the following six spaces (Shimamoto and Vanderwaart 2005;Curtis and Steiner 2006): the sphere, the torus, the sphere with two, three, or four handles, the disjoint union of two tori. Now we apply the method of parametrization of the last section which also works for non-equilateral pentagons.…”