@article{13188,
  abstract     = {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.
We 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.},
  author       = {Hafner, Christian and Bickel, Bernd},
  issn         = {1557-7368},
  journal      = {ACM Transactions on Graphics},
  keywords     = {Computer Graphics, Computational Design, Computational Geometry, Shape Modeling},
  number       = {5},
  publisher    = {Association for Computing Machinery},
  title        = {{The design space of Kirchhoff rods}},
  doi          = {10.1145/3606033},
  volume       = {42},
  year         = {2023},
}

@article{9817,
  abstract     = {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.},
  author       = {Hafner, Christian and Bickel, Bernd},
  issn         = {1557-7368},
  journal      = {ACM Transactions on Graphics},
  keywords     = {Computing methodologies, shape modeling, modeling and simulation, theory of computation, computational geometry, mathematics of computing, mathematical optimization},
  location     = {Virtual},
  number       = {4},
  publisher    = {Association for Computing Machinery},
  title        = {{The design space of plane elastic curves}},
  doi          = {10.1145/3450626.3459800},
  volume       = {40},
  year         = {2021},
}

@inproceedings{4097,
  abstract     = {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.},
  author       = {Edelsbrunner, Herbert and Guibas, Leonidas and Pach, János and Pollack, Richard and Seidel, Raimund and Sharir, Micha},
  booktitle    = {15th International Colloquium on Automata, Languages and Programming},
  isbn         = {978-3-540-19488-0},
  keywords     = {line segment, computational geometry, Jordan curve, cell decomposition, vertical tangency},
  location     = {Tampere, Finland},
  pages        = {214 -- 229},
  publisher    = {Springer},
  title        = {{Arrangements of curves in the plane - topology, combinatorics, and algorithms}},
  doi          = {10.1007/3-540-19488-6_118},
  volume       = {317},
  year         = {1988},
}