Abstract
Questions about lines in space arise frequently as subproblems in three-dimensional computational geometry. In this paper we study a number of fundamental combinatorial and algorithmic problems involving arrangements of n lines in three-dimensional space. Our main results include: 1. A tight Θ(n2) bound on the maximum combinatorial description complexity of the set of all oriented lines that have specified orientations relative to the n given lines. 2. A similar bound of Θ(n3) for the complexity of the set of all lines passing above the n given lines. 3. A preprocessing procedure using O(n2+ε) time and storage, for any ε > 0, that builds a structure supporting O(logn)-time queries for testing if a line lies above all the given lines. 4. An algorithm that tests the "towering property" in O(n4/3+ε) time, for any ε > 0: do n given red lines lie all above n given blue lines? The tools used to obtain these and other results include Plücker coordinates for lines in space and ε-nets for various geometric range spaces.
Original language | English (US) |
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Pages (from-to) | 428-447 |
Number of pages | 20 |
Journal | Algorithmica (New York) |
Volume | 15 |
Issue number | 5 |
DOIs | |
State | Published - 1996 |
Externally published | Yes |
All Science Journal Classification (ASJC) codes
- General Computer Science
- Applied Mathematics
- Computer Science Applications
Keywords
- Computational geometry
- Lines in space
- Plücker coordinates
- ε-Nets