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.

中文翻译：

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刚性位置少

由于 Gluck-Dehn-Alexandrov-Cauchy，对 $2$-球体进行三角测量的图在 $3$-空间中一般是刚性的。我们证明在 $3$-空间中存在一个 \emph{finite} 子集 $A$，因此每个图 $G$ 的顶点可以映射到 $A$ 中，从而使 $G$ 的嵌入结果无限小刚性。这个断言扩展到任何固定紧密连接表面的三角剖分，其中在 $A$ 大小上获得的上限随着属的增加而增加。断言失败，即不存在这样的有限 $A$，对于所有在 $3$-空间甚至平面中一般刚性的图的更大族。