标题： 二阶预应力稳定性和多面体曲面的三阶刚性 显示英文标题

标题： Second-order prestress stability and third-order rigidity of polyhedral surfaces

摘要： 基于能量的定义为几何约束系统中$n$阶刚度和柔性的正确定义提供了可行的解决方案。 将基于能量的局部刚度分析应用于多面曲面，我们推导出用于测试二阶预应力稳定性的新准则。 此外，我们讨论了测试三阶刚度的局限性，除非在刚度矩阵的零空间是一维的特殊情况。 尽管我们的主要研究对象是多面曲面，但本文中开发的理论和方法在一定条件下适用于一般的几何约束系统。 显示英文摘要

摘要： The energy-based definition provides a viable resolution to the longstanding confusion on the proper definition of $n$-th order rigidity and flexibility in geometric constraint systems. Applying an energy-based local rigidity analysis to polyhedral surfaces, we derive new criterion for testing second-order prestress stability. Additionally, we discuss the limitations of testing third-order rigidity, except in the special case where the null space of the rigidity matrix is one-dimensional. Although our primary focus is on polyhedral surfaces, the theory and methods developed in this article apply to general geometric constraint systems under certain conditions.