[{"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).","year":"2023","_id":"13188","abstract":[{"lang":"eng","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."}],"date_updated":"2024-03-25T23:30:26Z","oa_version":"Submitted Version","type":"journal_article","month":"09","volume":42,"file_date_updated":"2023-07-04T08:11:28Z","date_created":"2023-07-04T07:41:30Z","status":"public","external_id":{"isi":["001086833300010"]},"related_material":{"record":[{"relation":"part_of_dissertation","id":"12897","status":"public"}]},"intvolume":"        42","citation":{"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>","apa":"Hafner, C., &#38; Bickel, B. (2023). The design space of Kirchhoff rods. <i>ACM Transactions on Graphics</i>. 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Bickel, ACM Transactions on Graphics 42 (2023)."},"publication_status":"published","oa":1,"has_accepted_license":"1","date_published":"2023-09-20T00:00:00Z","ddc":["516"],"acknowledged_ssus":[{"_id":"M-Shop"}],"publisher":"Association for Computing Machinery","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","department":[{"_id":"BeBi"}],"publication":"ACM Transactions on Graphics","article_processing_charge":"No","ec_funded":1,"article_type":"original","author":[{"full_name":"Hafner, Christian","id":"400429CC-F248-11E8-B48F-1D18A9856A87","first_name":"Christian","last_name":"Hafner"},{"first_name":"Bernd","last_name":"Bickel","full_name":"Bickel, 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optimization"],"conference":{"end_date":"2021-08-13","start_date":"2021-08-09","location":"Virtual","name":"SIGGRAF: Special Interest Group on Computer Graphics and Interactive Techniques"},"isi":1,"project":[{"grant_number":"715767","call_identifier":"H2020","_id":"24F9549A-B435-11E9-9278-68D0E5697425","name":"MATERIALIZABLE: Intelligent fabrication-oriented Computational Design and Modeling"}],"file_date_updated":"2021-10-18T10:42:22Z","date_created":"2021-08-08T22:01:26Z","volume":40,"oa_version":"Published Version","month":"07","type":"journal_article","date_updated":"2024-03-25T23:30:26Z","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"}],"_id":"9817","year":"2021","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","date_published":"2021-07-19T00:00:00Z","ddc":["516"],"has_accepted_license":"1","publication_status":"published","oa":1,"citation":{"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>.","ista":"Hafner C, Bickel B. 2021. The design space of plane elastic curves. ACM Transactions on Graphics. 40(4), 126.","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>","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).","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.","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>."},"related_material":{"record":[{"relation":"dissertation_contains","status":"public","id":"12897"}],"link":[{"relation":"press_release","url":"https://ist.ac.at/en/news/designing-with-elastic-structures/","description":"News on IST Website"}]},"intvolume":"        40","external_id":{"isi":["000674930900091"]},"status":"public"},{"publication_identifier":{"isbn":["978-3-540-19488-0"]},"quality_controlled":"1","doi":"10.1007/3-540-19488-6_118","language":[{"iso":"eng"}],"conference":{"start_date":"1988-07-11","name":"ICALP: Automata, Languages and Programming","location":"Tampere, Finland","end_date":"1988-07-15"},"keyword":["line segment","computational geometry","Jordan curve","cell decomposition","vertical tangency"],"day":"01","author":[{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","first_name":"Herbert"},{"first_name":"Leonidas","last_name":"Guibas","full_name":"Guibas, Leonidas"},{"last_name":"Pach","first_name":"János","full_name":"Pach, János"},{"full_name":"Pollack, Richard","first_name":"Richard","last_name":"Pollack"},{"first_name":"Raimund","last_name":"Seidel","full_name":"Seidel, Raimund"},{"full_name":"Sharir, Micha","first_name":"Micha","last_name":"Sharir"}],"publist_id":"2028","title":"Arrangements of curves in the plane - topology, combinatorics, and algorithms","publisher":"Springer","user_id":"ea97e931-d5af-11eb-85d4-e6957dddbf17","scopus_import":"1","article_processing_charge":"No","publication":"15th International Colloquium on Automata, Languages and Programming","publication_status":"published","main_file_link":[{"url":"https://link.springer.com/chapter/10.1007/3-540-19488-6_118"}],"date_published":"1988-01-01T00:00:00Z","status":"public","alternative_title":["LNCS"],"citation":{"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.","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>.","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.","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>.","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>"},"intvolume":"       317","extern":"1","month":"01","oa_version":"None","type":"conference","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"}],"date_updated":"2022-02-08T10:15:09Z","page":"214 - 229","date_created":"2018-12-11T12:06:55Z","volume":317,"year":"1988","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.","_id":"4097"}]