论文标题
柔性多面体中的零和循环
Zero-sum cycles in flexible polyhedra
论文作者
论文摘要
我们表明，如果具有三角形面的三维仿射空间中的多面体是柔性的，即可以连续变形以保持其脸的形状，那么一个边缘的长度总和为零，一旦将一旦适当地加权1和1和-1。我们通过基本组合考虑因素来做到这一点，这是通过将三维仿射空间作为四维投影空间中的四维仿射空间的众所周知的压缩而实现的。紧凑型与欧几里得公制有关，使我们能够使用一种简单的变性技术，将问题降低到其一维模拟，这是微不足道的。
We show that if a polyhedron in the three-dimensional affine space with triangular faces is flexible, i.e., can be continuously deformed preserving the shape of its faces, then there is a cycle of edges whose lengths sum up to zero once suitably weighted by 1 and -1. We do this via elementary combinatorial considerations, made possible by a well-known compactification of the three-dimensional affine space as a quadric in the four-dimensional projective space. The compactification is related to the Euclidean metric, and allows us to use a simple degeneration technique that reduces the problem to its one-dimensional analogue, which is trivial to solve.
