New results on stabbing segments with a polygon

New Results on Stabbing Segments with a Polygon

We consider a natural variation of the concept of stabbing a set of segments with a simple polygon: a segment s is stabbed by a simple polygon P if at least one endpoint of s is contained in P, and a segment set S is stabbed by P if P stabs every element of S. Given a segment set S, we study the problem of finding a simple polygon P stabbing S in a way that some measure of P (such as area or pe...

Stabbing Segments with Rectilinear Objects

Given a set S of n line segments in the plane, we say that a region R ⊆ R is a stabber for S ifR contains exactly one endpoint of each segment of S. In this paper we provide optimal or near-optimal algorithms for reporting all combinatorially different stabbers for several shapes of stabbers. Specifically, we consider the case in which the stabber can be described as the intersection of axis-pa...

Stabbing Information of a Simple Polygon

Stabbing Convex Polygons with a Segment or a Polygon

Let O = {O1, . . . , Om} be a set of m convex polygons in R 2 with a total of n vertices, and let B be another convex k-gon. A placement of B, any congruent copy of B (without reflection), is called free if B does not intersect the interior of any polygon in O at this placement. A placement z of B is called critical if B forms three “distinct” contacts with O at z. Let φ(B,O) be the number of f...

3 Stabbing segments

In this paper, we answer the question: Given a collection of compact sets, can one decide in polynomial time whether there exists a convex body whose boundary intersects every set in the collection? We prove that already when the sets are segments in the plane, deciding existence of a convex stabber is NP-complete. On the positive side, we give a polynomial-time algorithm (in the full paper) to...