RUI: Folding and Unfolding Polytopes

这个项目主要关注的是算法方面的问题：给定一个简单的多边形，它可以折叠形成一个封闭的凸多面体（polytope）吗？虽然现在有大量的几何定理证明了折叠多面体的存在性和唯一性，但这些证明并不容易产生算法。找到这样的算法是本研究的主要目标。特别是，在折叠问题的几个变体中，给出更多的输入信息可能导致更容易的重构算法。多面体的展开也将被考虑。对于重要的应用程序来说，寻找不重叠的展开是一个有趣的计算问题。展开与多面体在平面上的滚动有关，这一概念会导致新的问题，也许会对多面体上最短路径的一些老问题有所启发。

This project's primary concern is with algorithmic aspects of the question: Given a simple polygon, can it be folded to form a closed convex polyhedron (a polytope)? Although there now exists a substantial body of geometry theorems that establish existence and uniqueness of the folded polytope, the proofs do not easily yield algorithms. Finding such algorithms is a major goal of this research. In particular, several variants of the folding problem in which more input information is given may lead to easier reconstruction algorithms. Unfoldings of polytopes will also be considered. Finding unfoldings that do not overlap is an interesting computational problem with important applications. Unfoldings are related to rollings of polytopes on a plane, a concept that leads to new problems, and perhaps will shed light on some old problems concerning shortest paths on polytopes.