Topological data analysis (TDA) provides tools for computing geometric and topological
information about spaces from a finite sample of points. We present an adaptive algorithm
for finding provably dense samples of points on real algebraic varieties given a set of
defining polynomials for use as input to TDA. The algorithm utilizes methods from numerical
algebraic geometry to give formal guarantees about the density of the sampling, and also
employs geometric heuristics to reduce the size of the sample. As TDA methods consume …

… § Pick a point and find the critical points of the minimal distance equations with the variety.
§ Record sample points, plus exclusion zone around these new sample points and the test
point. … Record sample points, plus exclusion zone around these new sample points and
the test point. § Pick another test point do the same. Repeat until …