The Number of Lines Tangent to Arbitrary Convex Polyhedra in 3D

Hervé Brönnimann (1) , Olivier Devillers (2) , Vida Dujmovic (3) , Hazel Everett (4) , Marc Glisse (4) , Xavier Goaoc (4) , Sylvain Lazard (4) , Hyeon-Suk Na (5) , Sue Whitesides (3)

1 Department of Computer and Information Science
2 GEOMETRICA - Geometric computing
3 School of Computer Science [Ottawa]
4 ISA - Models, algorithms and geometry for computer graphics and vision
5 School of Computing - Soongsil University, Séoul

We prove that the lines tangent to four possibly intersecting polytopes in $R3$ with $n$ edges in total form $\Theta(n^2)$ connected components. In the generic case, each connected component is a single line, but our result still holds for arbitrary degenerate scenes. This improves the previously known $O(n^3\log n)$ bound by Agarwal~\cite{A94}. More generally, we show that a set of $k$ polytopes with a total of $n$ edges admits, in the worse case, $\Theta(n^2k^2)$ connected components of lines tangent to any four of these polytopes.

Mots clés: géométrie algoritmique; visibilité 3d; 3d visibility; computational geometry

Domaines: Géométrie algorithmique [cs.CG]

Hervé Brönnimann, Olivier Devillers, Vida Dujmovic, Hazel Everett, Marc Glisse, et al.. The Number of Lines Tangent to Arbitrary Convex Polyhedra in 3D. Proceedings of the 20th Annual Symposium on Computational Geometry, Jun 2004, Brooklyn, NY, United States. pp.46 - 55, ⟨10.1145/997817.997827⟩. ⟨inria-00103995⟩