Book contents
- Frontmatter
- Contents
- Preface
- Translator's preface
- Acknowledgments
- Part I Algorithmic tools
- Part II Convex hulls
- Part III Triangulations
- Part IV Arrangements
- Chapter 14 Arrangements of hyperplanes
- Chapter 15 Arrangements of line segments in the plane
- Chapter 16 Arrangements of triangles
- Part V Voronoi diagrams
- References
- Notation
- Index
Chapter 15 - Arrangements of line segments in the plane
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Preface
- Translator's preface
- Acknowledgments
- Part I Algorithmic tools
- Part II Convex hulls
- Part III Triangulations
- Part IV Arrangements
- Chapter 14 Arrangements of hyperplanes
- Chapter 15 Arrangements of line segments in the plane
- Chapter 16 Arrangements of triangles
- Part V Voronoi diagrams
- References
- Notation
- Index
Summary
In an arrangement of n lines in the plane, all the cells are convex and thus have complexity O(n). Moreover, given a point A, the cell in the arrangement that contains A can be computed in time Θ(n log n): indeed, the problem reduces to computing the intersection of n half-planes bounded by the lines and containing A (see theorem 7.1.10).
In this chapter, we study arrangements of line segments in the plane. Consider a set S of n line segments in the plane. The arrangement of S includes cells, edges, and vertices of the planar subdivision of the plane induced by S, and their incidence relationships.
Computing the arrangement of S can be achieved in time O(n log n + k) where k is the number of intersection points (see sections 3.3 and 5.3.2, and theorem 5.2.5). All the pairs of segments may intersect, so in the worst case we have k = Ω(n2).
For a few applications, only a cell in this arrangement is needed. This is notably the case in robotics, for a polygonal robot moving amidst polygonal obstacles by translation (see exercise 15.6). The reachable positions are characterized by lying in a single cell of the arrangement of those line segments that correspond to the set of positions of the robot when a vertex of the robot slides along the edge of an obstacle, or when the edge of a robot maintains contact with an obstacle at a point.
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- Algorithmic Geometry , pp. 352 - 372Publisher: Cambridge University PressPrint publication year: 1998