  The Delaunay triangulation (DT) is one of the most common and useful
triangulations of point sets $P$ in the plane. DT is not unique when $P$ is
degenerate, specifically when it contains quadruples of co-circular points. One
way to achieve uniqueness is by applying a small (or infinitesimal)
perturbation to $P$.
  We consider a specific perturbation of such degenerate sets, in which the
coordinates of each point are independently perturbed by normally distributed
small quantities, and investigate the effect of such perturbations on the DT of
the set. We focus on two special configurations, where (1) the points of $P$
form a uniform grid, and (2) the points of $P$ are vertices of a regular
polygon.
  We present interesting (and sometimes surprising) empirical findings and
properties of the perturbed DTs for these cases, and give theoretical
explanations to some of them.
