  Topological data analysis (TDA) provides a growing body of tools for
computing geometric and topological information about spaces from a finite
sample of points. We present a new adaptive algorithm for finding provably
dense samples of points on real algebraic varieties given a set of defining
polynomials. The algorithm utilizes methods from numerical algebraic geometry
to give formal guarantees about the density of the sampling and it also employs
geometric heuristics to reduce the size of the sample. As TDA methods consume
significant computational resources that scale poorly in the number of sample
points, our sampling minimization makes applying TDA methods more feasible. We
provide a software package that implements the algorithm and also demonstrate
the implementation with several examples.
