On the Complexity of Sets of Free Lines and Line Segments Among Balls in Three Dimensions

Marc Glisse 1 Sylvain Lazard 2
1 GEOMETRICA - Geometric computing
CRISAM - Inria Sophia Antipolis - Méditerranée , Inria Saclay - Ile de France
2 VEGAS - Effective Geometric Algorithms for Surfaces and Visibility
INRIA Lorraine, LORIA - Laboratoire Lorrain de Recherche en Informatique et ses Applications
Abstract : We present two new fundamental lower bounds on the worst-case combinatorial complexity of sets of free lines and sets of maximal free line segments in the presence of balls in three dimensions. We first prove that the set of maximal non-occluded line segments among $n$ disjoint \emph{unit} balls has complexity $\Omega(n^4)$, which matches the trivial $O(n^4)$ upper bound. This improves the trivial $\Omega(n^2)$ bound and also the $\Omega(n^3)$ lower bound for the restricted setting of arbitrary-size balls [Devillers and Ramos, 2001]. This result settles, negatively, the natural conjecture that this set of line segments, or, equivalently, the visibility complex, has smaller worst-case complexity for disjoint fat objects than for skinny triangles. We also prove an $\Omega(n^3)$ lower bound on the complexity of the set of non-occluded lines among $n$ balls of arbitrary radii, improving on the trivial $\Omega(n^2)$ bound. This new bound almost matches the recent $O(n^{3+\epsilon})$ upper bound [Rubin, 2010].
Document type :
Conference papers
Complete list of metadatas

Cited literature [22 references]  Display  Hide  Download

https://hal.inria.fr/inria-00442751
Contributor : Sylvain Lazard <>
Submitted on : Wednesday, March 17, 2010 - 4:46:46 PM
Last modification on : Saturday, January 27, 2018 - 1:30:56 AM
Long-term archiving on : Wednesday, November 30, 2016 - 3:15:24 PM

File

HAL.pdf
Files produced by the author(s)

Identifiers

  • HAL Id : inria-00442751, version 2

Collections

Citation

Marc Glisse, Sylvain Lazard. On the Complexity of Sets of Free Lines and Line Segments Among Balls in Three Dimensions. 26th annual symposium on Computational geometry - SoCG 2010, Jun 2010, Snowbird, Utah, United States. ⟨inria-00442751v2⟩

Share

Metrics

Record views

431

Files downloads

196