硬盘簇的内蕴几何
Intrinsic Geometry of Hard Disk Clusters
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中文总结 AI 辅助
本文研究硬盘簇凸包周长最小化问题，建立任意有限大小硬盘簇的计算方法，解决了五圆盘情况，给出最小周长及三类解的性质，区分刚性等特性。
中文摘要 AI 辅助
将n个相同的硬币放在一张桌子上，任意两个都不重叠。哪种排列能使该簇的凸包的周长最小？尽管该问题表述简单，但目前仅能解决四个圆盘的情况。我们建立了任意有限大小硬盘簇的计算方法，给出了周长最小化的临界条件、一阶下降检验和二阶谱判据。一个核心难点在于周长公式会随凸包的组合结构变化，而可容许的一阶几何会随实际接触情况变化。我们的方法遵循如下原则：实际的几何结构本质上同时决定了泛函的局部形式和可容许的运动。作为应用，本文迈出了超越四个圆盘的第一步，给出了五个圆盘情况的解。最小周长为10+2π，仅在三个实际类中达到。其中两个类允许保周长的柔性变形，维数分别为1和2，而第三个类相对于刚体运动是刚性的。一阶理论提供了剪枝准则，简化的可容许空间及其内蕴海森矩阵可区分刚性、二阶不稳定性和周长平坦退化。
英文摘要
Put \(n\) identical coins on a table with no two overlapping. Which arrangement makes the perimeter of the convex hull of the cluster as small as possible? Despite its elementary statement, the solution of this problem is known only up to four disks. We produce a calculus for hard disk clusters of arbitrary finite size, providing class criticality conditions, first order descent tests, and second order spectral criteria for perimeter minimisation. A central difficulty is that the perimeter formula changes with the hull combinatorics, while the admissible first order geometry changes with the realised contacts. Our approach is guided by the principle that the realised geometry intrinsically determines both the local form of the functional and the admissible motions. As an application, this article takes the first step beyond four disks by providing a solution for the five disk case. The minimum perimeter is \(10+2π\), attained in exactly three realised classes. Two admit perimeter preserving flexes, of dimensions one and two, while the third is rigid modulo rigid motions. The first order theory provides pruning criteria, and the reduced admissible space together with its intrinsic Hessian distinguish rigidity, second order instability, and perimeter flat degeneracy.
