Recognizing Weakly Simple Polygons

Detecting Weakly Simple Polygons

5 A closed curve in the plane is weakly simple if it is the limit (in the Fréchet metric) 6 of a sequence of simple closed curves. We describe an algorithm to determine whether 7 a closed walk of length n in a simple plane graph is weakly simple in O(n log n) time, 8 improving an earlier O(n3)-time algorithm of Cortese et al. [Discrete Math. 2009]. As 9 an immediate corollary, we obtain the fir...

Reconstruction of Weakly Simple Polygons from their Edges

Given n line segments in the plane, do they form the edge set of a weakly simple polygon; that is, can the segment endpoints be perturbed by at most ε, for any ε > 0, to obtain a simple polygon? While the analogous question for simple polygons can easily be answered in O(n logn) time, we show that it is NP-complete for weakly simple polygons. We give O(n)-time algorithms in two special cases: w...

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. Toussaint [10] introduced these types of polyhedra as genera...

Recognizing S-Star Polygons

We consider the problem of recognizing star-polygons under staircase visibility (s-visibility). We show that the s-visibility polygon from a point inside a simple orthogonal polygon of n sides can be computed in O(n) time. When the polygon contains holes the algorithm runs in O(n log n) time, which we prove to be optimal by linear time reduction from sorting. We present an algorithm for computi...

On Recognizing and Characterizing Visibility Graphs of Simple Polygons