刚性，位置少
Rigidity with few locations
摘要：
在$3$ -空间中，三角化$2$ -球面的图是一般刚性的，这是由于Gluck-Dehn-Alexandrov-Cauchy的结果.我们证明了在$3$ -空间中存在一个\emph{finite}子集$A$，使得每个图$G$的顶点都可以映射到$A$中，从而使得$G$的嵌入是无穷小刚性的.这一论断扩展到任何固定的紧连通曲面的三角剖分，其中上界上得到的$A$的大小与属增加。断言失败，即没有这样的有限$A$存在，为更大的家庭的所有图是一般刚性的$3$ -空间，甚至在平面上。
英文摘要：
Graphs triangulating the $2$-sphere are generically rigid in $3$-space, due to Gluck-Dehn-Alexandrov-Cauchy. We show there is a \emph{finite} subset $A$ in $3$-space so that the vertices of each graph $G$ as above can be mapped into $A$ to make the resulted embedding of $G$ infinitesimally rigid. This assertion extends to the triangulations of any fixed compact connected surface, where the upper bound obtained on the size of $A$ increases with the genus. The assertion fails, namely no such finite $A$ exists, for the larger family of all graphs that are generically rigid in $3$-space and even in the plane.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
