Draining a polygon—or—rolling a ball out of a polygon

Abstract

We introduce the problem of draining water (or balls representing water drops) out of a punctured polygon (or a polyhedron) by rotating the shape. For 2D polygons, we obtain combinatorial bounds on the number of holes needed, both for arbitrary polygons and for special classes of polygons. We detail an O(n 2 log n) algorithm that finds the minimum number of holes needed for a given polygon, and argue that the complexity remains polynomial for polyhedra in 3D. We make a start at characterizing the 1-drainable shapes, those that only need one hole.

Pool segmentation for predicting water traps

“…Bose and Toussaint proposed an algorithm to find an orientation for a gravity casting mold that minimizes the number of venting holes that need to be added to allow air to escape to insure a complete fill [4]. Aloupis et al solved a 2D rotational draining problem for a closed polygon and a trapped single particle inside of the polygon, proposing an algorithm to find how many holes must be punctured to "drain" the particle [1]. For industrial applications of cleaning, on the other hand, we typically do not have the option of modifying the part by adding venting or draining holes to eliminate air or water traps.…”

Collecting a swarm in a grid environment using shared, global inputs

“…robot in a known workspace. This is similar to work on draining a polygon [19], or localizing a blind robot [11], [12], but with discrete inputs. In Section III, for the small particle problem we present an optimal collection algorithm, Alg.…”