Variable Radii Poisson - Disk Sampling extended version

We introduce three natural and well-defined generalizations of maximal Poisson-disk sampling. The first is to decouple the disk-free (inhibition) radius from the maximality (coverage) radius. Selecting a smaller inhibition radius than the coverage radius yields samples which mix advantages of Poisson-disk and uniform-random samplings. The second generalization yields hierarchical samplings, by scaling inhibition and coverage radii by an abstract parameter, e.g. time. The third generalization is to allow the radii to vary spatially, according to a formally characterized sizing function. We state bounds on edge lengths and angles in a Delaunay triangulation of the points, dependent on the ratio of inhibition to coverage radii, or the sizing function’s Lipschitz constant. Hierarchical samplings have distributions similar to those created directly. 1 Maximal Poisson-disk Sampling A sampling is a set of ordered points taken from a domain at random. Each point is the center of a disk that precludes additional points inside it, but points are otherwise chosen uniformly. The sampling is maximal if the entire domain is covered by disks. Together these define maximal Poisson-disk sampling (MPS), a.k.a. the limit distribution of the Matérn second process [18]. More formally, a sampling X = (xi) n i=1, xi ∈ Ω satisfies the inhibition or empty disk property if ∀i < j ≤ n, |xi − xj | ≥ r. (1) The set of uncovered points is defined to be S(X) = {y ∈ Ω : |y− xi| ≥ r, i = 1..n}. (2) A sampling X is maximal if S(X) is empty: