RUI: Configuration Spaces of Flexible Polyhedral Surfaces

想象一个由许多金属板组成的3D表面，这些金属板通过铰链沿沿着连接。这样一个表面是否可以从一个大圆顶开始，然后弯曲成一个紧凑的形状，小到足以装进火箭？这样一个表面的弯曲可以被控制，作为机器人机制的一部分吗？解决这些问题是这个项目的目标。PI将开发新的工具，以建立一般3D柔性多面体表面（可能看起来像一个圆顶，具有正曲率，或马鞍，具有负曲率）和折纸之间的强大连接，折纸是从平坦的零曲率纸折叠而成的。近年来，折纸折痕的折叠和展开在工程和物理学中的应用得到了大量的研究。将数学工具从折纸到灵活的多面体表面可以为建筑，机器人和外太空结构设计的实际应用开辟领域。此外，PI还将针对学生、教育工作者和公众举办研讨会和讲座，利用折纸的普及性，提高人们对STEM及其与艺术的交叉点的兴趣。本项目将研究并开发三种新工具，以建立柔性多面体表面与折纸之间的联系。第一个是一个新发现的对偶关系的顶点之间的多面体表面是椭圆形（具有正离散曲率）和双曲（具有负曲率）。PI将证明这种4度的对偶顶点在运动学上彼此等价（具有相同的运动学方程），以及与一个4度平面可折叠折纸顶点族等价，其运动学非常容易理解。第二个问题是建立一般折纸顶点折叠与平面折叠折纸顶点折叠之间的双射。第三个是找到一个几何解释，为什么它是如此有用的参数化的角度在每个铰链的一个灵活的多面体表面与正切的半角;这样的参数化往往线性化的配置空间的灵活的多面体顶点，但很少有人知道为什么。用于实现这些目标的技术将包括离散微分几何工具，如高斯地图和新的折纸工具，如中点正常轴的刚性折纸顶点和投影的折纸顶点到高维foldings.This奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。

Imagine a 3D surface made of many metal plates joined along their sides by hinges. Could such a surface start as a large dome and be flexed into a compact shape small enough to fit into a rocket? Could the flexing of such a surface be controlled to act as part of a robotics mechanism? Answering questions like these is the goal of this project. PI will develop new tools to establish a strong connection between general 3D flexible polyhedral surfaces (which might look like a dome, with positive curvature, or a saddle, with negative curvature) and origami, which is folded from flat, zero-curvature paper. The folding and unfolding of origami crease patterns has been studied heavily in recent years for applications in engineering and physics. Bringing mathematical tools from origami to flexible polyhedral surfaces could open up the field for practical applications in architecture, robotics, and structure designs for outer space. In addition, the PI will organize workshops and lectures on the topic of this project for students, educators, and the general public, leveraging the popularity of origami to increase interest in STEM and its intersections with art.This project will investigate and develop three new tools to establish connections between flexible polyhedral surfaces and origami. The first is a newly-discovered dual relationship between vertices in a polyhedral surface that are elliptic (have positive discrete curvature) and hyperbolic (with negative curvature). PI will prove that such dual vertices of degree 4 are kinematically equivalent to each other (have the same kinematic equations) as well as to a family of degree-4 flat-foldable origami vertices, whose kinematics are very well-understood. The second is to establish a bijection between foldings of general origami vertices and flat-foldable origami vertices. The third is to find a geometric explanation for why it is so useful to parameterize the angles at each hinge of a flexible polyhedral surface with the tangent of the half angle; such parameterizations often linearize the configuration space of flexible polyhedral vertices, but little is known as to why. The techniques used to achieve these goals will include discrete differential geometry tools like the Gauss map and new tools from origami like the midpoint normal axes of a rigid origami vertex and projections of origami vertices into higher dimensional foldings.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.