Fix a dimension d and graph H, with n vertices and m edges. Let p be a configuration of n points in R^d.

Then we can measure the configuration, mod the Euclidean group, by recording the squared length between each

point pair associated with an edge of H. When H is generically globally rigid in d-dimensions, then this measurement

map is an almost everywhere injective map from R^{nd}/E(d) to R^m. In this talk, I will discuss the general question of how

we can create fully injective maps from R^{nd}/G to R^m where G is some group and m is roughly 2nd. These maps will not be based on graphs.

This is work with Nadav Dym.