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

该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面件组装而不会产生材料应力，称为分段可展几何。这些将推动先进3D制造新兴行业的新发展：不仅可以从胶合板或金属板等丰富的薄平板材料中轻松切割可展几何形状，而且还可以提供一种新颖的几何方法来规划刀具路径，提高通过计算机控制的侧铣制造形状的效率和精度。 这样的工艺通过降低成本、增加制造形式的复杂性以及使以前只能通过手工实现的任务自动化（例如，可展开形式的机器人折叠）。 一个基本的问题，解决的研究是自动逼近的任意曲面的少量可开发件-目前这个过程必须进行费力的专家工程师和设计师，严重限制了范围和影响的可开发制造过程。 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，离散微分几何试图从离散算法的角度重新表述经典几何知识。 该领域提供了现代几何理论和需要结合数据和计算的工业/应用程序之间的重要联系，在该项目中培训的学生将有能力为3D制造做出贡献。该项目建立在光滑和离散微分几何的基础上：而不是仅仅将离散网格视为数值近似，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）可展性本身并不是离散可展性的充分表征，通常会导致讨厌的“起皱”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，这是服装设计中经常利用的一个事实。 这些意见导致两个主要的推力，即（i）表示离散的可展性，自然避免起皱，保证存在离散的“统治线”，和（ii）有效的算法，可展曲面设计的基础上稀疏的曲率描述沿着关键特征曲线。 一个交叉主题是制造的物理考虑，例如，简单几何模型和与物理制品的产生相关的材料/本构特性之间的转换。

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.