RUI: Configuration Spaces of Flexible Polyhedral Surfaces

想象一个由许多金属板组成的三维表面，这些金属板通过铰链沿其侧面连接在一起。这样的表面能否从一个大圆顶开始，然后弯曲成一个紧凑的形状，小到足以装进火箭？这种表面的弯曲是否可以作为机器人机制的一部分加以控制？回答这些问题是这个项目的目标。PI将开发新的工具，在一般的3D柔性多面体表面（可能看起来像一个圆顶，具有正曲率，或一个马鞍，具有负曲率）和折纸（由平坦的零曲率纸折叠而成）之间建立牢固的联系。近年来，折纸折痕图的折叠和展开在工程和物理领域得到了广泛的研究。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将为学生、教育工作者和公众举办有关该项目主题的研讨会和讲座，利用折纸的普及来增加对STEM及其与艺术交叉的兴趣。该项目将研究和开发三种新的工具来建立柔性多面体表面和折纸之间的联系。第一个是新发现的多面体曲面上椭圆（具有正离散曲率）和双曲（具有负曲率）顶点之间的对偶关系。PI将证明这样的4次对偶顶点在运动上是等价的（具有相同的运动方程），并且与一组4次可平折的折纸顶点是等价的，这些顶点的运动学是非常容易理解的。二是建立一般折纸顶点与可平折折纸顶点之间的对射。第三是找到一个几何解释，为什么用半角的正切来参数化一个柔性多面体表面的每个铰链的角度是如此有用；这种参数化常常使柔性多面体顶点的构型空间线性化，但其原因却鲜为人知。用于实现这些目标的技术将包括离散微分几何工具，如高斯图，以及折纸中的新工具，如刚性折纸顶点的中点法向轴和折纸顶点在高维折叠中的投影。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.