[{"_id":"17203","abstract":[{"lang":"eng","text":"The behavior of a rigid body primarily depends on its mass moments, which consist of the mass, center of mass, and moments of inertia. It is possible to manipulate these quantities without altering the geometric appearance of an object by introducing cavities in its interior. Algorithms that find cavities of suitable shapes and sizes have enabled the computational design of spinning tops, yo-yos, wheels, buoys, and statically balanced objects. Previous work is based, for example, on topology optimization on voxel grids, which introduces a large number of optimization variables and box constraints, or offset surface computation, which cannot guarantee that solutions to a feasible problem will always be found.\r\n\r\nIn this work, we provide a mathematical analysis of constrained topology optimization problems that depend only on mass moments. This class of problems covers, among others, all applications mentioned above. Our main result is to show that no matter the outer shape of the rigid body to be optimized or the optimization objective and constraints considered, the optimal solution always features a quadric-shaped interface between material and cavities. This proves that optimal interfaces are always ellipsoids, hyperboloids, paraboloids, or one of a few degenerate cases, such as planes.\r\n\r\nThis insight lets us replace a difficult topology optimization problem with a provably equivalent non-linear equation system in a small number (<10) of variables, which represent the coefficients of the quadric. This system can be solved in a few seconds for most examples, provides insights into the geometric structure of many specific applications, and lets us describe their solution properties. Finally, our method integrates seamlessly into modern fabrication workflows because our solutions are analytical surfaces that are native to the CAD domain."}],"intvolume":"        43","status":"public","publication_status":"published","file":[{"date_updated":"2024-07-05T12:05:17Z","creator":"chafner","success":1,"date_created":"2024-07-05T12:05:17Z","file_name":"sif-final.pdf","relation":"main_file","content_type":"application/pdf","access_level":"open_access","file_id":"17204","checksum":"0dc9f5a6422b8a49a79026900f349ee5","file_size":7225150},{"checksum":"cde433c6a40688d5f1187fb5721f6f94","file_size":397262,"access_level":"open_access","file_id":"17205","file_name":"sif-supp-final.pdf","date_created":"2024-07-05T12:06:03Z","relation":"supplementary_material","content_type":"application/pdf","date_updated":"2024-07-05T12:06:03Z","creator":"chafner"},{"date_created":"2024-07-17T09:29:13Z","file_name":"sif-video-final.mp4","content_type":"video/mp4","relation":"supplementary_material","date_updated":"2024-07-17T09:29:13Z","creator":"chafner","checksum":"c0457a09c2ab9a1c2935c995dcc84907","file_size":170001305,"title":"Submission Video","access_level":"open_access","file_id":"17276"}],"volume":43,"day":"01","keyword":["Topology Optimization","Mass Moments","Computational Geometry"],"type":"journal_article","article_processing_charge":"Yes (via OA deal)","publication":"Transactions on Graphics","issue":"4","publisher":"Association for Computing Machinery","citation":{"apa":"Hafner, C., Ly, M., &#38; Wojtan, C. (2024). Spin-it faster: Quadrics solve all topology optimization problems that depend only on mass moments. <i>Transactions on Graphics</i>. Denver, Colorado: Association for Computing Machinery. <a href=\"https://doi.org/10.1145/3658194\">https://doi.org/10.1145/3658194</a>","chicago":"Hafner, Christian, Mickaël Ly, and Chris Wojtan. “Spin-It Faster: Quadrics Solve All Topology Optimization Problems That Depend Only on Mass Moments.” <i>Transactions on Graphics</i>. Association for Computing Machinery, 2024. <a href=\"https://doi.org/10.1145/3658194\">https://doi.org/10.1145/3658194</a>.","ama":"Hafner C, Ly M, Wojtan C. Spin-it faster: Quadrics solve all topology optimization problems that depend only on mass moments. <i>Transactions on Graphics</i>. 2024;43(4). doi:<a href=\"https://doi.org/10.1145/3658194\">10.1145/3658194</a>","ista":"Hafner C, Ly M, Wojtan C. 2024. Spin-it faster: Quadrics solve all topology optimization problems that depend only on mass moments. Transactions on Graphics. 43(4), 78.","mla":"Hafner, Christian, et al. “Spin-It Faster: Quadrics Solve All Topology Optimization Problems That Depend Only on Mass Moments.” <i>Transactions on Graphics</i>, vol. 43, no. 4, 78, Association for Computing Machinery, 2024, doi:<a href=\"https://doi.org/10.1145/3658194\">10.1145/3658194</a>.","short":"C. Hafner, M. Ly, C. Wojtan, Transactions on Graphics 43 (2024).","ieee":"C. Hafner, M. Ly, and C. Wojtan, “Spin-it faster: Quadrics solve all topology optimization problems that depend only on mass moments,” <i>Transactions on Graphics</i>, vol. 43, no. 4. Association for Computing Machinery, 2024."},"oa":1,"quality_controlled":"1","corr_author":"1","article_type":"original","date_created":"2024-07-05T12:08:57Z","year":"2024","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","file_date_updated":"2024-07-17T09:29:13Z","title":"Spin-it faster: Quadrics solve all topology optimization problems that depend only on mass moments","date_updated":"2025-09-08T08:29:09Z","project":[{"_id":"34bc2376-11ca-11ed-8bc3-9a3b3961a088","name":"Computational Discovery of Numerical Algorithms for Animation and Simulation of Natural Phenomena","grant_number":"101045083"}],"author":[{"first_name":"Christian","full_name":"Hafner, Christian","last_name":"Hafner","id":"400429CC-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Ly","id":"6340d7f0-b48d-11eb-b10d-b7487e71d9f1","full_name":"Ly, Mickaël","first_name":"Mickaël"},{"id":"3C61F1D2-F248-11E8-B48F-1D18A9856A87","full_name":"Wojtan, Christopher J","last_name":"Wojtan","first_name":"Christopher J","orcid":"0000-0001-6646-5546"}],"date_published":"2024-07-01T00:00:00Z","ddc":["516"],"oa_version":"Published Version","scopus_import":"1","has_accepted_license":"1","month":"07","external_id":{"isi":["001289270900045"]},"language":[{"iso":"eng"}],"conference":{"location":"Denver, Colorado","end_date":"2024-08-01","start_date":"2024-07-28"},"acknowledgement":"We thank Gianmarco Cherchi for his help in tailoring the Mesh Booleans code for this project, Stefan Jeschke for his help with the photographs, Malina Strugaru and Aleksei Kalinov for their help with the samples, and the anonymous reviewers as well as the members of the ISTA Visual Computing Group for their feedback. This project was funded in part by the European Research Council (ERC Consolidator Grant 101045083 CoDiNA).","publication_identifier":{"issn":["0730-0301"],"eissn":["1557-7368"]},"doi":"10.1145/3658194","department":[{"_id":"ChWo"}],"isi":1,"article_number":"78"},{"has_accepted_license":"1","oa_version":"Submitted Version","scopus_import":"1","month":"09","external_id":{"isi":["001086833300010"]},"acknowledged_ssus":[{"_id":"M-Shop"}],"language":[{"iso":"eng"}],"date_updated":"2026-04-26T22:30:03Z","project":[{"name":"MATERIALIZABLE: Intelligent fabrication-oriented Computational Design and Modeling","call_identifier":"H2020","_id":"24F9549A-B435-11E9-9278-68D0E5697425","grant_number":"715767"}],"author":[{"first_name":"Christian","full_name":"Hafner, Christian","id":"400429CC-F248-11E8-B48F-1D18A9856A87","last_name":"Hafner"},{"full_name":"Bickel, Bernd","id":"49876194-F248-11E8-B48F-1D18A9856A87","last_name":"Bickel","first_name":"Bernd","orcid":"0000-0001-6511-9385"}],"ddc":["516"],"date_published":"2023-09-20T00:00:00Z","isi":1,"department":[{"_id":"BeBi"}],"article_number":"171","acknowledgement":"We thank the anonymous reviewers for their generous feedback, and Julian Fischer for his help in proving Proposition 1. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 715767).","publication_identifier":{"issn":["0730-0301"],"eissn":["1557-7368"]},"doi":"10.1145/3606033","day":"20","keyword":["Computer Graphics","Computational Design","Computational Geometry","Shape Modeling"],"volume":42,"type":"journal_article","article_processing_charge":"No","publisher":"Association for Computing Machinery","issue":"5","publication":"ACM Transactions on Graphics","status":"public","_id":"13188","abstract":[{"text":"The Kirchhoff rod model describes the bending and twisting of slender elastic rods in three dimensions, and has been widely studied to enable the prediction of how a rod will deform, given its geometry and boundary conditions. In this work, we study a number of inverse problems with the goal of computing the geometry of a straight rod that will automatically deform to match a curved target shape after attaching its endpoints to a support structure. Our solution lets us finely control the static equilibrium state of a rod by varying the cross-sectional profiles along its length.\r\nWe also show that the set of physically realizable equilibrium states admits a concise geometric description in terms of linear line complexes, which leads to very efficient computational design algorithms. Implemented in an interactive software tool, they allow us to convert three-dimensional hand-drawn spline curves to elastic rods, and give feedback about the feasibility and practicality of a design in real time. We demonstrate the efficacy of our method by designing and manufacturing several physical prototypes with applications to interior design and soft robotics.","lang":"eng"}],"intvolume":"        42","file":[{"file_size":19635168,"checksum":"4954c1cfa487725bc156dcfec872478a","access_level":"open_access","file_id":"13194","date_created":"2023-07-04T08:11:28Z","file_name":"kirchhoff-rods.pdf","content_type":"application/pdf","relation":"main_file","date_updated":"2023-07-04T08:11:28Z","creator":"chafner","success":1},{"file_size":420909,"checksum":"79c9975fbc82ff71f1767331d2204cca","file_id":"13190","access_level":"open_access","title":"Supplemental Material with Proofs","content_type":"application/pdf","relation":"supplementary_material","file_name":"supp-main.pdf","date_created":"2023-07-04T07:46:28Z","creator":"chafner","date_updated":"2023-07-04T07:46:28Z"},{"file_id":"13191","title":"Cheat Sheet for Notation","access_level":"open_access","checksum":"4ab647e4f03c711e1e6a5fc1eb8684db","file_size":430086,"creator":"chafner","date_updated":"2023-07-04T07:46:30Z","content_type":"application/pdf","relation":"supplementary_material","file_name":"supp-cheat.pdf","date_created":"2023-07-04T07:46:30Z"},{"relation":"supplementary_material","content_type":"video/mp4","file_name":"kirchhoff-video-final.mp4","date_created":"2023-07-04T07:46:39Z","creator":"chafner","date_updated":"2023-07-04T07:46:39Z","checksum":"c0fd9a57d012046de90c185ffa904b76","file_size":268088064,"file_id":"13192","access_level":"open_access","title":"Supplemental Video"},{"date_updated":"2023-07-04T07:47:10Z","creator":"chafner","file_name":"matlab-submission.zip","date_created":"2023-07-04T07:47:10Z","content_type":"application/x-zip-compressed","relation":"supplementary_material","access_level":"open_access","title":"Matlab Source Code with Example","file_id":"13193","checksum":"71b00712b489ada2cd9815910ee180a9","file_size":25790}],"publication_status":"published","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","related_material":{"record":[{"id":"12897","relation":"part_of_dissertation","status":"public"}]},"ec_funded":1,"title":"The design space of Kirchhoff rods","file_date_updated":"2023-07-04T08:11:28Z","citation":{"mla":"Hafner, Christian, and Bernd Bickel. “The Design Space of Kirchhoff Rods.” <i>ACM Transactions on Graphics</i>, vol. 42, no. 5, 171, Association for Computing Machinery, 2023, doi:<a href=\"https://doi.org/10.1145/3606033\">10.1145/3606033</a>.","short":"C. Hafner, B. Bickel, ACM Transactions on Graphics 42 (2023).","ieee":"C. Hafner and B. Bickel, “The design space of Kirchhoff rods,” <i>ACM Transactions on Graphics</i>, vol. 42, no. 5. Association for Computing Machinery, 2023.","apa":"Hafner, C., &#38; Bickel, B. (2023). The design space of Kirchhoff rods. <i>ACM Transactions on Graphics</i>. Association for Computing Machinery. <a href=\"https://doi.org/10.1145/3606033\">https://doi.org/10.1145/3606033</a>","chicago":"Hafner, Christian, and Bernd Bickel. “The Design Space of Kirchhoff Rods.” <i>ACM Transactions on Graphics</i>. Association for Computing Machinery, 2023. <a href=\"https://doi.org/10.1145/3606033\">https://doi.org/10.1145/3606033</a>.","ama":"Hafner C, Bickel B. The design space of Kirchhoff rods. <i>ACM Transactions on Graphics</i>. 2023;42(5). doi:<a href=\"https://doi.org/10.1145/3606033\">10.1145/3606033</a>","ista":"Hafner C, Bickel B. 2023. The design space of Kirchhoff rods. ACM Transactions on Graphics. 42(5), 171."},"oa":1,"article_type":"original","quality_controlled":"1","corr_author":"1","year":"2023","date_created":"2023-07-04T07:41:30Z"},{"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png"},"isi":1,"department":[{"_id":"BeBi"}],"article_number":"126","acknowledgement":"We thank the anonymous reviewers for their generous feedback, and Michal Piovarči for his help in producing the supplemental video. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 715767).\r\n","publication_identifier":{"eissn":["1557-7368"],"issn":["0730-0301"]},"doi":"10.1145/3450626.3459800","has_accepted_license":"1","scopus_import":"1","oa_version":"Published Version","month":"07","external_id":{"isi":["000674930900091"]},"conference":{"end_date":"2021-08-13","start_date":"2021-08-09","location":"Virtual","name":"SIGGRAF: Special Interest Group on Computer Graphics and Interactive Techniques"},"language":[{"iso":"eng"}],"date_updated":"2026-04-26T22:30:03Z","project":[{"grant_number":"715767","call_identifier":"H2020","_id":"24F9549A-B435-11E9-9278-68D0E5697425","name":"MATERIALIZABLE: Intelligent fabrication-oriented Computational Design and Modeling"}],"author":[{"first_name":"Christian","full_name":"Hafner, Christian","last_name":"Hafner","id":"400429CC-F248-11E8-B48F-1D18A9856A87"},{"orcid":"0000-0001-6511-9385","first_name":"Bernd","last_name":"Bickel","id":"49876194-F248-11E8-B48F-1D18A9856A87","full_name":"Bickel, Bernd"}],"ddc":["516"],"date_published":"2021-07-19T00:00:00Z","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","related_material":{"record":[{"relation":"dissertation_contains","id":"12897","status":"public"}],"link":[{"url":"https://ist.ac.at/en/news/designing-with-elastic-structures/","relation":"press_release","description":"News on IST Website"}]},"ec_funded":1,"title":"The design space of plane elastic curves","file_date_updated":"2021-10-18T10:42:22Z","citation":{"chicago":"Hafner, Christian, and Bernd Bickel. “The Design Space of Plane Elastic Curves.” <i>ACM Transactions on Graphics</i>. Association for Computing Machinery, 2021. <a href=\"https://doi.org/10.1145/3450626.3459800\">https://doi.org/10.1145/3450626.3459800</a>.","apa":"Hafner, C., &#38; Bickel, B. (2021). The design space of plane elastic curves. <i>ACM Transactions on Graphics</i>. Virtual: Association for Computing Machinery. <a href=\"https://doi.org/10.1145/3450626.3459800\">https://doi.org/10.1145/3450626.3459800</a>","ista":"Hafner C, Bickel B. 2021. The design space of plane elastic curves. ACM Transactions on Graphics. 40(4), 126.","ama":"Hafner C, Bickel B. The design space of plane elastic curves. <i>ACM Transactions on Graphics</i>. 2021;40(4). doi:<a href=\"https://doi.org/10.1145/3450626.3459800\">10.1145/3450626.3459800</a>","short":"C. Hafner, B. Bickel, ACM Transactions on Graphics 40 (2021).","mla":"Hafner, Christian, and Bernd Bickel. “The Design Space of Plane Elastic Curves.” <i>ACM Transactions on Graphics</i>, vol. 40, no. 4, 126, Association for Computing Machinery, 2021, doi:<a href=\"https://doi.org/10.1145/3450626.3459800\">10.1145/3450626.3459800</a>.","ieee":"C. Hafner and B. Bickel, “The design space of plane elastic curves,” <i>ACM Transactions on Graphics</i>, vol. 40, no. 4. Association for Computing Machinery, 2021."},"oa":1,"article_type":"original","quality_controlled":"1","year":"2021","date_created":"2021-08-08T22:01:26Z","day":"19","keyword":["Computing methodologies","shape modeling","modeling and simulation","theory of computation","computational geometry","mathematics of computing","mathematical optimization"],"volume":40,"type":"journal_article","article_processing_charge":"No","publisher":"Association for Computing Machinery","issue":"4","publication":"ACM Transactions on Graphics","status":"public","_id":"9817","abstract":[{"text":"Elastic bending of initially flat slender elements allows the realization and economic fabrication of intriguing curved shapes. In this work, we derive an intuitive but rigorous geometric characterization of the design space of plane elastic rods with variable stiffness. It enables designers to determine which shapes are physically viable with active bending by visual inspection alone. Building on these insights, we propose a method for efficiently designing the geometry of a flat elastic rod that realizes a target equilibrium curve, which only requires solving a linear program. We implement this method in an interactive computational design tool that gives feedback about the feasibility of a design, and computes the geometry of the structural elements necessary to realize it within an instant. The tool also offers an iterative optimization routine that improves the fabricability of a model while modifying it as little as possible. In addition, we use our geometric characterization to derive an algorithm for analyzing and recovering the stability of elastic curves that would otherwise snap out of their unstable equilibrium shapes by buckling. We show the efficacy of our approach by designing and manufacturing several physical models that are assembled from flat elements.","lang":"eng"}],"intvolume":"        40","file":[{"date_updated":"2021-10-18T10:42:15Z","creator":"chafner","success":1,"date_created":"2021-10-18T10:42:15Z","file_name":"elastic-curves-paper.pdf","content_type":"application/pdf","relation":"main_file","access_level":"open_access","file_id":"10150","file_size":17064290,"checksum":"7e5d08ce46b0451b3102eacd3d00f85f"},{"checksum":"0088643478be7c01a703b5b10767348f","file_size":547156,"file_id":"10151","access_level":"open_access","content_type":"application/pdf","relation":"supplementary_material","date_created":"2021-10-18T10:42:22Z","file_name":"elastic-curves-supp.pdf","creator":"chafner","date_updated":"2021-10-18T10:42:22Z"}],"publication_status":"published"},{"acknowledgement":"Work on this paper by the first author has been supported by Amoco Fnd. Fac. Dev. Comput. Sci. 1-6-44862 and by the National Science Foundation under grant CCR-8714566. Work on this paper by the third and sixth authors has been supported by Office of Naval Research Grant N00014-82-K-0381, by National Science Foundation Grant No. NSF-DCR-83-20085, by grants from the Digital Equipment Corporation, and the IBM Corporation. Work by the sixth author has also been supported by a research grant from the NCRD — the Israeli National Council for Research and Development. Work by the fourth author has been supported by National Science Foundation Grant DMS-8501947.","publication_identifier":{"isbn":["978-3-540-19488-0"]},"doi":"10.1007/3-540-19488-6_118","oa_version":"None","scopus_import":"1","month":"01","language":[{"iso":"eng"}],"conference":{"name":"ICALP: Automata, Languages and Programming","location":"Tampere, Finland","start_date":"1988-07-11","end_date":"1988-07-15"},"date_updated":"2022-02-08T10:15:09Z","author":[{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","orcid":"0000-0002-9823-6833","first_name":"Herbert"},{"last_name":"Guibas","full_name":"Guibas, Leonidas","first_name":"Leonidas"},{"first_name":"János","last_name":"Pach","full_name":"Pach, János"},{"full_name":"Pollack, Richard","last_name":"Pollack","first_name":"Richard"},{"first_name":"Raimund","last_name":"Seidel","full_name":"Seidel, Raimund"},{"full_name":"Sharir, Micha","last_name":"Sharir","first_name":"Micha"}],"date_published":"1988-01-01T00:00:00Z","alternative_title":["LNCS"],"user_id":"ea97e931-d5af-11eb-85d4-e6957dddbf17","title":"Arrangements of curves in the plane - topology, combinatorics, and algorithms","publist_id":"2028","citation":{"ista":"Edelsbrunner H, Guibas L, Pach J, Pollack R, Seidel R, Sharir M. 1988. Arrangements of curves in the plane - topology, combinatorics, and algorithms. 15th International Colloquium on Automata, Languages and Programming. ICALP: Automata, Languages and Programming, LNCS, vol. 317, 214–229.","ama":"Edelsbrunner H, Guibas L, Pach J, Pollack R, Seidel R, Sharir M. Arrangements of curves in the plane - topology, combinatorics, and algorithms. In: <i>15th International Colloquium on Automata, Languages and Programming</i>. Vol 317. Springer; 1988:214-229. doi:<a href=\"https://doi.org/10.1007/3-540-19488-6_118\">10.1007/3-540-19488-6_118</a>","chicago":"Edelsbrunner, Herbert, Leonidas Guibas, János Pach, Richard Pollack, Raimund Seidel, and Micha Sharir. “Arrangements of Curves in the Plane - Topology, Combinatorics, and Algorithms.” In <i>15th International Colloquium on Automata, Languages and Programming</i>, 317:214–29. Springer, 1988. <a href=\"https://doi.org/10.1007/3-540-19488-6_118\">https://doi.org/10.1007/3-540-19488-6_118</a>.","apa":"Edelsbrunner, H., Guibas, L., Pach, J., Pollack, R., Seidel, R., &#38; Sharir, M. (1988). Arrangements of curves in the plane - topology, combinatorics, and algorithms. In <i>15th International Colloquium on Automata, Languages and Programming</i> (Vol. 317, pp. 214–229). Tampere, Finland: Springer. <a href=\"https://doi.org/10.1007/3-540-19488-6_118\">https://doi.org/10.1007/3-540-19488-6_118</a>","mla":"Edelsbrunner, Herbert, et al. “Arrangements of Curves in the Plane - Topology, Combinatorics, and Algorithms.” <i>15th International Colloquium on Automata, Languages and Programming</i>, vol. 317, Springer, 1988, pp. 214–29, doi:<a href=\"https://doi.org/10.1007/3-540-19488-6_118\">10.1007/3-540-19488-6_118</a>.","short":"H. Edelsbrunner, L. Guibas, J. Pach, R. Pollack, R. Seidel, M. Sharir, in:, 15th International Colloquium on Automata, Languages and Programming, Springer, 1988, pp. 214–229.","ieee":"H. Edelsbrunner, L. Guibas, J. Pach, R. Pollack, R. Seidel, and M. Sharir, “Arrangements of curves in the plane - topology, combinatorics, and algorithms,” in <i>15th International Colloquium on Automata, Languages and Programming</i>, Tampere, Finland, 1988, vol. 317, pp. 214–229."},"quality_controlled":"1","date_created":"2018-12-11T12:06:55Z","year":"1988","volume":317,"keyword":["line segment","computational geometry","Jordan curve","cell decomposition","vertical tangency"],"day":"01","page":"214 - 229","type":"conference","article_processing_charge":"No","extern":"1","publication":"15th International Colloquium on Automata, Languages and Programming","publisher":"Springer","main_file_link":[{"url":"https://link.springer.com/chapter/10.1007/3-540-19488-6_118"}],"_id":"4097","intvolume":"       317","abstract":[{"text":"Arrangements of curves in the plane are of fundamental significance in many problems of computational and combinatorial geometry (e.g. motion planning, algebraic cell decomposition, etc.). In this paper we study various topological and combinatorial properties of such arrangements under some mild assumptions on the shape of the curves, and develop basic tools for the construction, manipulation, and analysis of these arrangements. Our main results include a generalization of the zone theorem of [EOS], [CGL] to arrangements of curves (in which we show that the combinatorial complexity of the zone of a curve is nearly linear in the number of curves), and an application of (some weaker variant of) that theorem to obtain a nearly quadratic incremental algorithm for the construction of such arrangements.","lang":"eng"}],"status":"public","publication_status":"published"}]