The number of lines tangent to arbitrary convex polyhedra in 3D

We prove that the lines tangent to four possibly intersecting convex polyhedra in   3 with n edges in total form Θ(n2 ) 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 convex polyhedra with a total of n edges admits, in the worst case, Θ(n2 k2 ) connected components of (possibly occluded) lines tangent to any four of these polyhedra. We also show a lower bound of Ω(n2 k2 ) on the number of non-occluded maximal line segments tangent to any four of these k convex polyhedra. Categories and Subject Descriptors F.2.2 [Nonnumerical Algorithms and Problems]: Geometrical problems and computations; G.2.1 [Combinatorics]: Counting problems; I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling; I.3.7 [ThreeDimensional Graphics and Realism]: Visible line/surface algorithms General Terms Algorithms, Theory. Keywords Comput...

• Type: Conference
• Year: 2004
• Where: COMPGEOM
• Authors: Hervé Brönnimann, Olivier Devillers, Vida Dujmovic, Hazel Everett, Marc Glisse, Xavier Goaoc, Sylvain Lazard, Hyeon-Suk Na, Sue Whitesides
• Keywords: COMPGEOM 2004, Convex Polyhedra, Lines Tangent, Worst Case