AF: Small: Collaborative Research: Computational Representations for Design and Fabrication of Developable Surfaces

这个项目开发了三维形式的数学表示和计算机算法，这些形式可以从一组平板组装而不会产生材料应力，即所谓的分段可展几何。这些将推动先进3D制造这一新兴行业的新发展：不仅可以轻松地从胶合板或板材等丰富的薄平板材料中切割出可展开的几何图形，而且还可以提供一种新颖的几何路径规划方法，提高通过计算机控制的侧切加工制造形状的效率和精度。这种工艺通过降低成本、增加制作模板的复杂性以及自动化以前只能由人工完成的任务(例如，可展开表单的机器人折叠)，为美国的制造业提供了竞争优势。这项研究解决的一个基本问题是用少量可展开的零件自动逼近任意曲面-目前这一过程必须由专家工程师和设计师费力地进行，严重限制了可展开制造工艺的范围和影响。更广泛地说，可展几何的算法模型丰富了离散微分几何领域的基本理解，它试图从离散的、算法的观点重新表述经典几何知识。这一领域在现代几何理论和需要结合数据和计算的工业/应用应用之间提供了重要的联系，接受过该项目培训的学生将具备在3D制造方面做出贡献的良好准备。该项目建立在光滑和离散微分几何的基础上：不是将离散网格视为单纯的数值近似，统一的目标是开发直接编码分段可展几何最显著特征的数据结构。两个关键的观察结果是：(I)可展平性本身并不是离散可展性的充分表征，往往会导致令人讨厌的“褶皱”行为；(Ii)分段可展曲面的曲率完全由其面片边界的形状编码，这一事实经常被服装设计所利用。这些观察结果导致了两个主要的推力，即(I)离散可展性的表示，通过保证离散“统治线”的存在自然地避免皱缩，以及(Ii)基于沿关键特征曲线的曲率的稀疏描述的可展曲面设计的高效算法。一个贯穿各领域的主题是制造的物理考虑，例如，在简单几何模型和与物理人工制品的生产相关的材料/本构属性之间的转换。

This project develops mathematical representations and computer algorithms for three-dimensional forms that can be assembled from a collection of flat pieces without incurring material stress, known as piecewise developable geometry. These will drive new developments in the emerging industry of advanced 3D manufacturing: Not only can developable geometry can easily be cut from a rich array of thin flat materials like plywood or sheet metal, but it can also provide a novel geometric approach to tool path planning that improves the efficiency and accuracy of shapes fabricated via computer-controlled flank milling. Such processes offer a competitive advantage for manufacturing in the US by reducing cost, increasing complexity of fabricated forms, and automating tasks previously only achievable by hand (e.g., robotic folding of developable forms). A fundamental issue addressed by the research is automatic approximation of an arbitrary curved surface by a small number of developable pieces---at present this process must be carried out laboriously by expert engineers and designers, severely limiting the scope and impact of developable manufacturing processes. More broadly, algorithmic models of developable geometry enrich basic understanding in the area of discrete differential geometry, which seeks to reformulate classical geometric knowledge from a discrete, algorithmic point of view. This area provides a crucial link between modern geometric theory and industrial/applied applications that need to incorporate data and computation, and students trained in this project will be well-equipped to contribute in 3d manufacturing. The project builds on foundations from smooth and discrete differential geometry: rather than view discrete meshes as mere numerical approximations, the unifying goal is to develop data structures that directly encode the most salient features of piecewise developable geometry. Two key observations are that (i) flattenability alone is not a sufficient characterization for discrete developability, often leading to nasty "crumpling" behavior and (ii) the curvature of a piecewise developable surface is encoded entirely by the shape of its patch boundaries, a fact often exploited in garment design. These observations lead to two primary thrusts, namely (i) representations for discrete developability that naturally avoid crumpling by guaranteeing the existence of discrete "ruling lines," and (ii) efficient algorithms for developable surface design based on sparse descriptions of curvature along critical feature curves. A cross-cutting theme is physical considerations for fabrication, e.g., translation between simple geometric models and material/constitutive properties relevant to the production of physical artifacts.