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

Authors:
H. Brönnimann;O. Devillers;V. Dujmović;H. Everett;M. Glisse;X. Goaoc;S. Lazard;H.-S. Na;S. Whitesides
Affiliations:
Polytechnic University, NY;INRIA Sophia-Antipolis, Sophia-Antipolis, France;McGill University, Montreal, Canada;LORIA, Nancy, France;LORIA, Nancy, France;LORIA, Nancy, France;LORIA - INRIA Lorraine, Nancy, France;Soongsil University, Seoul, S. Korea;McGill University, Montreal, Canada
Venue:
SCG '04 Proceedings of the twentieth annual symposium on Computational geometry
Year:
2004

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Abstract

We prove that the lines tangent to four possibly intersecting convex polyhedra in ℝ3 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 kconvex polyhedra with a total of n edges admits, in the worst case, Θ(n2k2)connected components of (possibly occluded) lines tangent to any four of these polyhedra. We also show a lower bound of Ω(n2k2) on the number of non-occluded maximal line segments tangent to any four of these k convex polyhedra.