Rigidity with few locations

TL;DR

This paper proves that for graphs triangulating the 2-sphere and other surfaces, there exists a finite set of points in 3-space such that the vertices can be mapped into this set to achieve infinitesimal rigidity, extending known results.

Contribution

It establishes the existence of finite point sets in 3-space for embedding surface triangulation graphs to ensure infinitesimal rigidity, generalizing previous rigidity results.

Findings

Finite sets exist for surface triangulation graphs to achieve rigidity.

The size of these sets depends on the surface's genus.

No such finite set exists for all generically rigid graphs in 3-space or the plane.

Abstract

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.