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Many infinite families of polytopes are described by the direction of their facet normals. This is the data that becomes encoded in the "left hand side" constraint matrix $A$ of the family. It is interesting, from an experimental point of view, to consider all polytopes with a fixed constraint matrix $A$. In this talk, we will consider the theoretical framework for generating a representative collection of polytopes (one per combinatorial type) by studying a bijection first given by L. J. Billera, I.M. Gelf'and, and B. Sturmfels: There is a correspondence between the chambers of $cone(A)$ and the regular triangulations of the Gale transform of $A$.