AI 中文总结
研究保持棱长和面共面性的多面体变形，开发二阶理论及算法测试二阶刚性，在多面体类上验证优势，发现特殊案例，还研究棱长扰动及与刚性的联系，并以正十二面体为例进行案例研究。
AI 中文摘要
我们研究保持棱长和面共面性的多面体变形。由此产生了一种刚性概念，我们为此开发了二阶理论以及用于符号测试二阶刚性的有效算法。这些新工具的优势在几类多面体上得到证明，包括之前难以处理的正十二面体，我们发现它是刚性的。我们还发现一个测试的多面体避开了我们的技术，因此将为更高阶工具的未来发展提供一个简单测试案例。本文还包含对所谓棱长扰动的研究。我们指出刚性与用略有扰动的棱长实现多面体的能力之间的联系。为了在实践中探索这些联系，我们专门用一节对正十二面体进行另一个案例研究。我们研究其实现空间的局部行为，以探讨棱长扰动、奇点和一般全局刚性。
英文摘要
We study deformations of polytopes that preserve edge lengths and face coplanarities. This gives rise to a notion of rigidity, for which we develop a second-order theory together with an effective algorithm for testing second-order rigidity symbolically. The strength of these new tools is demonstrated on several polytope classes, including the previously intractable regular dodecahedron, which we find to be rigid. We also find that a single tested polytope evades our techniques and will therefore provide a simple test case for future developments of tools of even higher order. This paper also contains a study of so-called edge-length perturbations. We point out connections between rigidity and the ability to realize polytopes with slightly perturbed edge lengths. To explore these connections in practice, we dedicate a section to another case study of the regular dodecahedron. We investigate the local behavior of its realization space with a view towards edge-length perturbations, singularities and generic global rigidity.
