# Stabbing convex polygons with a segment or a polygon

Journal Article

Let O = {O1, . . . , Om} be a set of m convex polygons in ℝ2 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 at this placement. A placement z of B is called critical if B forms three "distinct" contacts with at z. Let be the number of free critical placements. A set of placements of B is called a stabbing set of if each polygon in intersects at least one placement of B in this set. We develop efficient Monte Carlo algorithms that compute a stabbing set of size h = O(h *logm), with high probability, where h * is the size of the optimal stabbing set of O. We also improve bounds on (B, O) for the following three cases, namely, (i) B is a line segment and the obstacles in are O pairwise-disjoint, (ii) B is a line segment and the obstacles in O may intersect (iii) B is a convex k-gon and the obstacles in O are disjoint, and use these improved bounds to analyze the running time of our stabbing-set algorithm. © 2008 Springer-Verlag Berlin Heidelberg.

### Full Text

### Duke Authors

### Cited Authors

- Agarwal, PK; Chen, DZ; Ganjugunte, SK; Misiołek, E; Sharir, M; Tang, K

### Published Date

- January 1, 2008

### Published In

### Volume / Issue

- 5193 LNCS /

### Start / End Page

- 52 - 63

### Electronic International Standard Serial Number (EISSN)

- 1611-3349

### International Standard Serial Number (ISSN)

- 0302-9743

### Digital Object Identifier (DOI)

- 10.1007/978-3-540-87744-8_5

### Citation Source

- Scopus