Published 2005
| Version v1
Report
On the Number of Maximal Free Line Segments Tangent to Arbitrary Three-dimensional Convex Polyhedra
Contributors
Others:
- Department of Computer and Information Science ; Polytechnic Institute of New York University
- Geometric computing (GEOMETRICA) ; Centre Inria d'Université Côte d'Azur (CRISAM) ; Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)
- School of computer science [Ottawa] (SCS) ; University of Ottawa [Ottawa]
- Effective Geometric Algorithms for Surfaces and Visibility (VEGAS) ; INRIA Lorraine ; Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Lorrain de Recherche en Informatique et ses Applications (LORIA) ; Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)
- School of Computing - Soongsil University, Séoul ; Soongsil University, Seoul
- INRIA
Description
We prove that the lines tangent to four possibly intersecting convex polyhedra in $ ^3$ with $n$ edges in total form $\Theta(n^2)$ connected components in the worst case. In the generic case, each connected component is a single line, but our result still holds for arbitrary degenerate scenes. More generally, we show that a set of $k$ possibly intersecting convex polyhedra with a total of $n$ edges admits, in the worst case, $\Theta(n^2k^2)$ connected components of maximal free line segments tangent to any four of the polytopes. This bound also holds for the number of connected components of possibly occluded lines tangent to any four of the polytopes.
Additional details
Identifiers
- URL
- https://inria.hal.science/inria-00071226
- URN
- urn:oai:HAL:inria-00071226v1
Origin repository
- Origin repository
- UNICA