{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,22]],"date-time":"2026-08-22T08:14:23Z","timestamp":1787386463174,"version":"build-2736575974"},"reference-count":25,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Comput."],"published-print":{"date-parts":[[2007,1]]},"abstract":"<jats:p>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.<\/jats:p>","DOI":"10.1137\/s0097539705447116","type":"journal-article","created":{"date-parts":[[2007,6,6]],"date-time":"2007-06-06T18:00:23Z","timestamp":1181152823000},"page":"522-551","source":"Crossref","is-referenced-by-count":11,"title":["Lines and Free Line Segments Tangent to Arbitrary Three-Dimensional Convex Polyhedra"],"prefix":"10.1137","volume":"37","author":[{"given":"Herv\u00e9","family":"Br\u00f6nnimann","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Olivier","family":"Devillers","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Vida","family":"Dujmovi\u0107","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hazel","family":"Everett","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Marc","family":"Glisse","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xavier","family":"Goaoc","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Sylvain","family":"Lazard","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hyeon-Suk","family":"Na","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Sue","family":"Whitesides","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2007,6,5]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1016\/0925-7721(94)90016-7"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1007\/s00454-005-1166-2"},{"key":"R3","doi-asserted-by":"crossref","unstructured":"P. K. Agarwal and M. Sharir,\n                      Ray shooting amidst convex polyhedra and polyhedral terrains\n                      in three dimensions, SIAM J. Comput., 25 (1996), pp. 100\u2013116.","DOI":"10.1137\/S0097539793244368"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1016\/S0925-7721(98)00020-0"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1142\/S0218195997000120"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1007\/BF01187019"},{"key":"R7","doi-asserted-by":"crossref","unstructured":"J. D. Boissonnat and M. Yvinec,\n                      Algorithmic Geometry\n                      , Cambridge University Press, Cambridge, UK, 1998.","DOI":"10.1017\/CBO9781139172998"},{"key":"R8","unstructured":"H. Br\u00f6nnimann, O. Devillers, V. Dujmovi\u0107, H. Everett, M. Glisse, X. Goaoc, S. Lazard, H.S. Na, and S. Whitesides,\n                      On the number of lines tangent to four convex polyhedra\n                      , in Proceedings of the 14th Annual Canadian Conference on Computational Geometry, Lethbridge, Canada, 2002, pp. 113\u2013117."},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1007\/s00454-005-1183-1"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1016\/S0097-8493(99)00093-X"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1016\/S0747-7171(89)80003-3"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1137\/S0097539702419662"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1016\/0304-3975(82)90120-7"},{"key":"R14","unstructured":"F. Durand,\n                      A Multidisciplinary Survey of Visibility\n                      , ACM SIGGRAPH Course Notes: Visibility, Problems, Techniques, and Applications, 2000."},{"key":"R15","doi-asserted-by":"crossref","unstructured":"F. Durand, G. Drettakis, and C. Puech,\n                      The visibility skeleton: A powerful and efficient multi-purpose global visibility tool\n                      , in Proceedings of the 24th Annual Conference on Computer Graphics and Interactive Techniques, 1997, pp. 89\u2013100.","DOI":"10.1145\/258734.258785"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1145\/508357.508362"},{"key":"R17","unstructured":"A. Efrat, L. J. Guibas, O. A. Hall-Holt, and L. Zhang,\n                      On incremental rendering of silhouette maps of a polyhedral scene\n                      , in Proceedings of the Eleventh ACM-SIAM Symposium on Discrete Algorithms, San Francisco, 2000, pp. 910\u2013917."},{"key":"R18","unstructured":"H. Everett, S. Lazard, W. Lenhart, J. Redburn, and L. Zhang,\n                      Predicates for line transversals in 3D\n                      , in Proceedings of the 18th Annual Canadian Conference on Computational Geometry, 2006, pp. 43\u201346."},{"key":"R19","unstructured":"X. Goaoc,\n                      Structures de visibilit\u00e9 globales: Taille, calcul et d\u00e9g\u00e9n\u00e9rescences\n                      , Ph.D. thesis, Universit\u00e9 Nancy 2, Nancy, France, 2004."},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1007\/BF02574383"},{"key":"R21","unstructured":"J. O'Rourke,\n                      Computational Geometry in C\n                      , 2nd ed., Cambridge University Press, Cambridge, UK, 1998."},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1007\/BF02574376"},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1142\/S0218195996000204"},{"key":"R24","unstructured":"M. Sharir and P. K. Agarwal,\n                      Davenport-Schinzel Sequences and Their Geometric Applications\n                      , Cambridge University Press, Cambridge, UK, 1995."},{"key":"R25","doi-asserted-by":"crossref","unstructured":"J. Stolfi,\n                      Oriented Projective Geometry: A Framework for Geometric Computations\n                      , Academic Press, New York, 1991.","DOI":"10.1016\/B978-0-12-672025-9.50004-6"}],"container-title":["SIAM Journal on Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/S0097539705447116","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:26:31Z","timestamp":1787336791000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/S0097539705447116"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,1]]},"references-count":25,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2007,1]]}},"alternative-id":["10.1137\/S0097539705447116"],"URL":"https:\/\/doi.org\/10.1137\/s0097539705447116","relation":{},"ISSN":["0097-5397","1095-7111"],"issn-type":[{"value":"0097-5397","type":"print"},{"value":"1095-7111","type":"electronic"}],"subject":[],"published":{"date-parts":[[2007,1]]}}}