Generalizing monotonicity: On recognizing special classes of polygons and polyhedra

A simple polyhedron is weakly-monotonic in direction d→ provided that the intersection of the polyhedron and any plane with normal d→ is simply-connected (i.e. empty, a point, a line-segment or a simple polygon). Furthermore, if the intersection is a convex set, then the polyhedron is said to be weakly-monotonic in the convex sense. Toussaint10 introduced these types of polyhedra as generalizations of the 2-dimensional notion of monotonicity. We study the following recognition problems: Given a simple n-vertex polyhedron in 3-dimensions, we present an O(n log n) time algorithm to determine if there exists a direction d→ such that when sweeping over the polyhedron with a plane in direction d→, the cross-section (or intersection) is a convex set. If we allow multiple convex polygons in the cross-section as opposed to a single convex polygon, then we provide a linear-time recognition algorithm. For simply-connected cross-sections (i.e. the cross-section is empty, a point, a line-segment or a simple polygon), we derive an O(n2) time recognition algorithm to determine if a direction d→ exists. We then study variations of monotonicity in 2-dimensions, i.e. for simple polygons. Given a simple n-vertex polygon P, we can determine whether or not a line ℓ can be swept over P in a continuous manner but with varying direction, such that any position of ℓ intersects P in at most two edges. We study two variants of the problem: one where the line is allowed to sweep over a portion of the polygon multiple times and one where it can sweep any portion of the polygon only once. Although the latter problem is slightly more complicated than the former since the line movements are restricted, our solutions to both problems run in O(n2) time