Motivated by visibility problems in three dimensions, we investigate the complexity and construction of the set of tangent lines in a scene of three-dimensional polyhedra. We prove that the set of lines tangent to four possibly intersecting convex polyhedra in $\mathbb{R}^3$ with a total of n edges consists of $\Theta(n^2)$ connected components in the worst case. In the generic case, each connected component is a single line, but our result still holds for arbitrarily 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 at least four polytopes. Furthermore, these bounds also hold for possibly occluded lines rather than maximal free line segments. Finally, we present an $O(n^2 k^2 \log n)$ time and $O(nk^2)$ space algorithm that, given a scene of k possibly intersecting convex polyhedra, computes all the minimal free line segments that are tangent to any four of the polytopes and are isolated transversals to the set of edges they intersect; in particular, we compute at least one line segment per connected component of tangent lines.

  • [1]  P. K. Agarwal, On stabbing lines for convex polyhedra in 3D, Comput. Geom., 4 (1994), pp. 177–189. CGOME6 0925-7721 CrossrefISIGoogle Scholar

  • [2]  P. K. Agarwal, B. Aronov, V. Koltun and , and M. Sharir, Lines avoiding unit balls in three dimensions, Discrete Comput. Geom., 34 (2005), pp. 231–250. DCGEER 0179-5376 CrossrefISIGoogle Scholar

  • [3]  P. K. Agarwal and M. Sharir, Ray shooting amidst convex polyhedra and polyhedral terrains in three dimensions, SIAM J. Comput., 25 (1996), pp. 100–116. Google Scholar

  • [4]  M. de Berg, H. Everett and , and L. J. Guibas, The union of moving polygonal pseudodiscs—Combinatorial bounds and applications, Comput. Geom., 11 (1998), pp. 69–81. CGOME6 0925-7721 CrossrefISIGoogle Scholar

  • [5]  M. de Berg, D. Halperin, M. Overmars and , and M. van Kreveld, Sparse arrangements and the number of views of polyhedral scenes, Internat. J. Comput. Geom. Appl., 7 (1997), pp. 175–195. IJCAEV 0218-1959 CrossrefISIGoogle Scholar

  • [6]  M. Bern, D. P. Dobkin, D. Eppstein and , and R. Grossman, Visibility with a moving point of view, Algorithmica, 11 (1994), pp. 360–378. ALGOEJ 0178-4617 CrossrefISIGoogle Scholar

  • [7]  J. D. Boissonnat and M. Yvinec, Algorithmic Geometry, Cambridge University Press, Cambridge, UK, 1998. Google Scholar

  • [8]  H. Brönnimann, O. Devillers, V. Dujmović, H. Everett, M. Glisse, X. Goaoc, S. Lazard, H.-S. Na, and S. Whitesides, On the number of lines tangent to four convex polyhedra, in Proceedings of the 14th Annual Canadian Conference on Computational Geometry, Lethbridge, Canada, 2002, pp. 113–117. Google Scholar

  • [9]  H. Brönnimann, H. Everett, S. Lazard, F. Sottile and , and S. Whitesides, Transversals to line segments in three-dimensional space, Discrete Comput. Geom., 34 (2005), pp. 381–390. DCGEER 0179-5376 CrossrefISIGoogle Scholar

  • [10]  F. S. Cho and  and D. Forsyth, Interactive ray tracing with the visibility complex, Computers and Graphics, 23 (1999), pp. 703–717. CrossrefISIGoogle Scholar

  • [11]  R. Cole and  and M. Sharir, Visibility problems for polyhedral terrains, J. Symbolic Comput., 7 (1989), pp. 11–30. JSYCEH 0747-7171 CrossrefISIGoogle Scholar

  • [12]  O. Devillers, V. Dujmović, H. Everett, X. Goaoc, S. Lazard, H.-S. Na and , and S. Petitjean, The expected number of 3D visibility events is linear, SIAM J. Comput., 32 (2003), pp. 1586–1620. SMJCAT 0097-5397 LinkISIGoogle Scholar

  • [13]  D. P. Dobkin and  and D. G. Kirkpatrick, Fast detection of polyhedral intersection, Theoret. Comput. Sci., 27 (1983), pp. 241–253. TCSCDI 0304-3975 CrossrefISIGoogle Scholar

  • [14]  F. Durand, A Multidisciplinary Survey of Visibility, ACM SIGGRAPH Course Notes: Visibility, Problems, Techniques, and Applications, 2000. Google Scholar

  • [15]  F. Durand, G. Drettakis, and C. Puech, The visibility skeleton: A powerful and efficient multi-purpose global visibility tool, in Proceedings of the 24th Annual Conference on Computer Graphics and Interactive Techniques, 1997, pp. 89–100. Google Scholar

  • [16]  F. Durand, G. Drettakis and , and C. Puech, The 3D visibility complex, ACM Trans. Graphics, 21 (2002), pp. 176–206. ATGRDF 0730-0301 CrossrefISIGoogle Scholar

  • [17]  A. Efrat, L. J. Guibas, O. A. Hall-Holt, and L. Zhang, On incremental rendering of silhouette maps of a polyhedral scene, in Proceedings of the Eleventh ACM-SIAM Symposium on Discrete Algorithms, San Francisco, 2000, pp. 910–917. Google Scholar

  • [18]  H. Everett, S. Lazard, W. Lenhart, J. Redburn, and L. Zhang, Predicates for line transversals in 3D, in Proceedings of the 18th Annual Canadian Conference on Computational Geometry, 2006, pp. 43–46. Google Scholar

  • [19]  X. Goaoc, Structures de visibilité globales: Taille, calcul et dégénérescences, Ph.D. thesis, Université Nancy 2, Nancy, France, 2004. Google Scholar

  • [20]  D. Halperin and  and M. Sharir, New bounds for lower envelopes in three dimensions, with applications to visbility in terrains, Discrete Comput. Geom., 12 (1994), pp. 313–326. DCGEER 0179-5376 CrossrefISIGoogle Scholar

  • [21]  J. O'Rourke, Computational Geometry in C, 2nd ed., Cambridge University Press, Cambridge, UK, 1998. Google Scholar

  • [22]  M. Pellegrini, On lines missing polyhedral sets in 3-space, Discrete Comput. Geom., 12 (1994), pp. 203–221. DCGEER 0179-5376 CrossrefISIGoogle Scholar

  • [23]  M. Pocchiola and  and G. Vegter, The visibility complex, Internat. J. Comput. Geom. Appl., 6 (1996), pp. 279–308. IJCAEV 0218-1959 CrossrefISIGoogle Scholar

  • [24]  M. Sharir and P. K. Agarwal, Davenport-Schinzel Sequences and Their Geometric Applications, Cambridge University Press, Cambridge, UK, 1995. Google Scholar

  • [25]  J. Stolfi, Oriented Projective Geometry: A Framework for Geometric Computations, Academic Press, New York, 1991. Google Scholar