CoBarS: Fast Reweighted Sampling for Polygon Spaces

Conformal barycenter sampling is a method for generating random samples from polygon spaces in R^d where the number of edges n and the edgelengths r₁, …, r_n are fixed. The basic idea is that a configuration of such a polygon (up to translation) may be described by a weighted point cloud w₁, …, w_n on S^d-1 where the weighted sum r₁ w₁ + … + r_n w_n = 0.

We can generate configurations of w_i by sampling them uniformly from the sphere and using an appropriate Möbius transformation to recenter them. This method introduces some sampling bias, but the appropriate weights to correct for the bias are computed in the paper (CoBarS: Fast Reweighted Sampling for Polygon Spaces in any Dimension, Jason Cantarella and Henrik Schumacher, 2023).

CoBarS is extremely fast (linear time in fixed dimension) and produces almost-uniform samples even without the weights. The method is implemented in two GitHub repositories:

• CoBarS (a header-only cpp library which implements the sampling method)
• CoBarSLink (a Mathematica interface for CoBarS)