RUI: Configuration Spaces of Flexible Polyhedral Surfaces

想象一个由许多金属板组成的3D表面，这些金属板的两侧由铰链连接。这样的表面可以从一个大圆顶开始，然后弯曲成足够小的紧凑形状，可以装进火箭吗？这种表面的弯曲能被控制成机器人机械的一部分吗？回答这样的问题是这个项目的目标。PI将开发新的工具，在常规3D柔性多面体曲面(可能看起来像圆顶，具有正曲率，或鞍形，具有负曲率)和折纸之间建立强大的连接，折纸是从平面的零曲率纸张折叠而成的。折纸折痕图案的折叠和展开在工程和物理中的应用近年来得到了广泛的研究。将数学工具从折纸到灵活的多面体表面，可以为建筑、机器人和外层空间结构设计的实际应用开辟领域。此外，PI将为学生、教育工作者和公众组织关于这个项目主题的工作坊和讲座，利用折纸的流行来提高人们对STEM及其与艺术的交叉的兴趣。这个项目将调查和开发三个新的工具，以建立柔性多面体表面和折纸之间的联系。第一种是新发现的多面体曲面中椭圆(具有正的离散曲率)和双曲线(具有负曲率)的顶点之间的对偶关系。PI将证明这样的4次对偶顶点在运动学上彼此等价(具有相同的运动学方程)以及一族4次可平折折纸顶点，其运动学非常容易理解。第二种方法是建立一般折纸顶点的折叠与可平折折纸顶点之间的双射。三是寻找几何解释，解释为什么用半角的切线将柔性多面体曲面的每个铰链处的角度参数化是如此有用；这种参数化通常使柔性多面体顶点的配置空间线性化，但对原因知之甚少。用来实现这些目标的技术将包括离散的微分几何工具，如高斯地图，以及来自折纸的新工具，如刚性折纸顶点的中点法线轴，以及将折纸顶点投影到更高维的折叠中。该奖项反映了NSF的法定使命，并已通过使用基金会的智力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.