The isomorphism problem of planar polygon spaces

Real projective space as a space of planar polygons

We prove that real projective space RP^{n-3} is homeomorphic to the space of all isometry classes of n-gons in the plane with one side of length n-2 and all other sides of length 1. This makes the topological complexity of real projective space more relevant to robotics.

二连通和三连通平面多边形上的最优 Reeb 图

Optimal Reeb graphs on two- and three-connected planar polygon

本文利用带标记顶点的有向 Reeb 图刻画一般位置平面多边形梯形分解的拓扑结构，分类单极小、单极大情形下的最优 Reeb 图，并给出六顶点二连通与九顶点三连通多边形的梯形映射结构。

平面C多边形的顶点分类

Vertex Classification of Planar C-polygons

论文研究了平面C多边形的顶点特性，证明了严格凸域C的C多边形的奇点边界点数量范围，并针对译移C多边形给出了更精确的范围。

Moduli Spaces of Planar Pentagonal Linkages: Combinatorial Description

Moduli spaces of planar polygonal linkages admit a cell structure which can be realized as a surgery on the permutohedron. We present a 3D visualization of the result of the surgery for all types of non-degenerate pentagonal linkages.

Extremal polygons in R^3

The oriented area function $A$ is (generically) a Morse function on the space of planar configurations of a polygonal linkage. We are lucky to have an easy description of its critical points as cyclic polygons and a simple formula for the Morse index of a critical point. However, for planar polygons, the function $A$ in many cases is not a perfect Morse function. In particular, for an equilateral pentagonal linkage it has one extra local maximum (except for the global maximum) and one extra local minimum. In the present paper we consider the space of 3D configurations of a polygonal linkage. F…

Oriented Area as a Morse Function on Polygon Spaces

We study polygon spaces arising from planar configurations of necklaces with some of the beads fixed and some of the beads sliding freely. These spaces include configuration spaces of flexible polygons and some other natural polygon spaces. We characterise critical points of the oriented area function in geometric terms and give a formula for the Morse indices. Thus we obtain a generalisation of isoperimetric theorems for polygons in the plane.