@inproceedings{22405,
  abstract     = {Computing the diameter of the intersection graphs of objects is a basic problem in computational geometry. Previous works showed that the complexity of computing the diameter mainly depends on the object types: for unit disks and squares in 2D, the problem is solvable in truly subquadratic time [Chan et al., 2025], while for other objects, including unit segments and equilateral triangles in 2D or unit balls and axis-parallel unit cubes in 3D, there is no truly subquadratic time algorithm under the Orthogonal Vector (OV) hypothesis [Bringmann et al., 2022]. 
We undertake a comprehensive study of computing the diameter of geometric intersection graphs for various types of objects. We discover many new irregularities, showing that the landscape is extremely nuanced: the source of hardness is a combination of the object type, the true diameter value, and how the objects intersect with each other. Our highlighted results for the 2D case include:  
1) The diameter of non-degenerate, axis-aligned line segments can be computed in truly subquadratic time. Previous hardness result [Bringmann et al., 2022] for line segments applies only to degenerate instances. On the other hand, for the degenerate case, we show that a truly subquadratic time algorithm exists when the true diameter is constant. 
2) An almost-linear-time algorithm for unit-square graphs of constant diameter. Previous algorithms [Duraj et al., 2024; Chan et al., 2025] rely on succinct representation assuming bounded VC-dimension; for such a strategy Ω(n^{7/4}) time is an inherent barrier. 
3) An Õ(n^{4/3})-time algorithm to decide if the diameter of a unit-disk graph is at most 2. This improves upon the recent algorithm with running time Õ(n^{2-1/9}) [Chan et al., 2025]. 
4) Deciding if the diameter of intersection graphs of fat triangles or line segments is at most 2 is truly subquadratic-hard under fine-grained complexity assumptions. Previous lower bounds [Bringmann et al., 2022] only hold when deciding if diameter is at most 3.  Our findings are presented in a pair of papers. This paper focuses solely on the 2D case, while the companion paper is devoted to higher-dimensional cases.},
  author       = {Chan, Timothy M. and Chang, Hsien-Chih and Gao, Jie and Kisfaludi-Bak, Sándor and Le, Hung and Zheng, Da Wei},
  booktitle    = {53rd International Colloquium on Automata, Languages, and Programming},
  issn         = {1868-8969},
  keywords     = {String graphs, Fine-grained complexity, Theory of computation → Computational geometry},
  location     = {Egham, United Kingdom},
  publisher    = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
  title        = {{Charting the landscape of diameter computation on geometric intersection graphs in the plane}},
  doi          = {10.4230/LIPICS.ICALP.2026.54},
  volume       = {374},
  year         = {2026},
}

@inproceedings{22299,
  abstract     = {The depth poset of a filtered Lefschetz complex reflects the dependencies between the cancellations of different shallow birth-death pairs. Using the fast algorithms for computing the depth poset in [Edelsbrunner et al., 2026] and for updating the persistence diagram under transpositions in [Cohen-Steiner et al., 2006], we give a complete case analysis of how transpositions of cells in the filter affect the depth poset. In addition, we present statistics on the depth poset for random point data and its sensitivity to the transpositions that occur in random straight-line homotopies.},
  author       = {Edelsbrunner, Herbert and Lipiński, Michał and Mrozek, Marian and Soriano Trigueros, Manuel and Zimin, Fedor},
  booktitle    = {42nd International Symposium on Computational Geometry},
  isbn         = {9783959774185},
  issn         = {1868-8969},
  keywords     = {Algebraic topology, Lefschetz complexes, persistent homology, vines and vineyards, birth-death pairs, shallow pairs, relations, partial orders, transpositions, Theory of computation → Computational geometry},
  location     = {New Brunswick, NJ, United States},
  publisher    = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
  title        = {{The depth poset under transpositions in the filter}},
  doi          = {10.4230/LIPICS.SOCG.2026.41},
  volume       = {367},
  year         = {2026},
}

@article{17203,
  abstract     = {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.

In 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.

This 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.},
  author       = {Hafner, Christian and Ly, Mickaël and Wojtan, Christopher J},
  issn         = {1557-7368},
  journal      = {Transactions on Graphics},
  keywords     = {Topology Optimization, Mass Moments, Computational Geometry},
  location     = {Denver, Colorado},
  number       = {4},
  publisher    = {Association for Computing Machinery},
  title        = {{Spin-it faster: Quadrics solve all topology optimization problems that depend only on mass moments}},
  doi          = {10.1145/3658194},
  volume       = {43},
  year         = {2024},
}

@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},
}

@article{3994,
  abstract     = {The body defined by a finite collection of disks is a subset of the plane bounded by a tangent continuous curve, which we call the skin. We give analytic formulas for the area, the perimeter, the area derivative, and the perimeter derivative of the body. Given the filtrations of the Delaunay triangulation and the Voronoi diagram of the disks, all formulas can be evaluated in time proportional to the number of disks.},
  author       = {Cheng, Ho and Edelsbrunner, Herbert},
  journal      = {Computational Geometry},
  keywords     = {Computational geometry, Differential geometry, Skin curves, Voronoi diagrams, Delaunay triangulations, Filtrations, Disks, Hyperbolas, Area, Perimeter, Derivatives},
  number       = {2},
  pages        = {173 -- 192},
  publisher    = {Elsevier},
  title        = {{Area, perimeter and derivatives of a skin curve}},
  doi          = {10.1016/S0925-7721(02)00124-4},
  volume       = {26},
  year         = {2003},
}

@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},
}