Rigidity with few locations

Graphs triangulating the 2-sphere are generically rigid in 3-space, due to Gluck-Dehn-Alexandrov-Cauchy. We show there is a 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.