---
_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."
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).
article_number: '78'
article_processing_charge: Yes (via OA deal)
article_type: original
author:
- first_name: Christian
  full_name: Hafner, Christian
  id: 400429CC-F248-11E8-B48F-1D18A9856A87
  last_name: Hafner
- first_name: Mickaël
  full_name: Ly, Mickaël
  id: 6340d7f0-b48d-11eb-b10d-b7487e71d9f1
  last_name: Ly
- first_name: Christopher J
  full_name: Wojtan, Christopher J
  id: 3C61F1D2-F248-11E8-B48F-1D18A9856A87
  last_name: Wojtan
  orcid: 0000-0001-6646-5546
citation:
  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>'
  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>.'
  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.'
  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).
conference:
  end_date: 2024-08-01
  location: Denver, Colorado
  start_date: 2024-07-28
corr_author: '1'
date_created: 2024-07-05T12:08:57Z
date_published: 2024-07-01T00:00:00Z
date_updated: 2025-09-08T08:29:09Z
day: '01'
ddc:
- '516'
department:
- _id: ChWo
doi: 10.1145/3658194
external_id:
  isi:
  - '001289270900045'
file:
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  file_size: 170001305
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  title: Submission Video
file_date_updated: 2024-07-17T09:29:13Z
has_accepted_license: '1'
intvolume: '        43'
isi: 1
issue: '4'
keyword:
- Topology Optimization
- Mass Moments
- Computational Geometry
language:
- iso: eng
month: '07'
oa: 1
oa_version: Published Version
project:
- _id: 34bc2376-11ca-11ed-8bc3-9a3b3961a088
  grant_number: '101045083'
  name: Computational Discovery of Numerical Algorithms for Animation and Simulation
    of Natural Phenomena
publication: Transactions on Graphics
publication_identifier:
  eissn:
  - 1557-7368
  issn:
  - 0730-0301
publication_status: published
publisher: Association for Computing Machinery
quality_controlled: '1'
scopus_import: '1'
status: public
title: 'Spin-it faster: Quadrics solve all topology optimization problems that depend
  only on mass moments'
type: journal_article
user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 43
year: '2024'
...
---
_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."
acknowledged_ssus:
- _id: M-Shop
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).
article_number: '171'
article_processing_charge: No
article_type: original
author:
- first_name: Christian
  full_name: Hafner, Christian
  id: 400429CC-F248-11E8-B48F-1D18A9856A87
  last_name: Hafner
- first_name: Bernd
  full_name: Bickel, Bernd
  id: 49876194-F248-11E8-B48F-1D18A9856A87
  last_name: Bickel
  orcid: 0000-0001-6511-9385
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>. 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>.
  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.
  ista: Hafner C, Bickel B. 2023. The design space of Kirchhoff rods. ACM Transactions
    on Graphics. 42(5), 171.
  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).
corr_author: '1'
date_created: 2023-07-04T07:41:30Z
date_published: 2023-09-20T00:00:00Z
date_updated: 2026-04-23T22:30:03Z
day: '20'
ddc:
- '516'
department:
- _id: BeBi
doi: 10.1145/3606033
ec_funded: 1
external_id:
  isi:
  - '001086833300010'
file:
- access_level: open_access
  checksum: 4954c1cfa487725bc156dcfec872478a
  content_type: application/pdf
  creator: chafner
  date_created: 2023-07-04T08:11:28Z
  date_updated: 2023-07-04T08:11:28Z
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  file_size: 19635168
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  creator: chafner
  date_created: 2023-07-04T07:46:28Z
  date_updated: 2023-07-04T07:46:28Z
  file_id: '13190'
  file_name: supp-main.pdf
  file_size: 420909
  relation: supplementary_material
  title: Supplemental Material with Proofs
- access_level: open_access
  checksum: 4ab647e4f03c711e1e6a5fc1eb8684db
  content_type: application/pdf
  creator: chafner
  date_created: 2023-07-04T07:46:30Z
  date_updated: 2023-07-04T07:46:30Z
  file_id: '13191'
  file_name: supp-cheat.pdf
  file_size: 430086
  relation: supplementary_material
  title: Cheat Sheet for Notation
- access_level: open_access
  checksum: c0fd9a57d012046de90c185ffa904b76
  content_type: video/mp4
  creator: chafner
  date_created: 2023-07-04T07:46:39Z
  date_updated: 2023-07-04T07:46:39Z
  file_id: '13192'
  file_name: kirchhoff-video-final.mp4
  file_size: 268088064
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  title: Supplemental Video
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  checksum: 71b00712b489ada2cd9815910ee180a9
  content_type: application/x-zip-compressed
  creator: chafner
  date_created: 2023-07-04T07:47:10Z
  date_updated: 2023-07-04T07:47:10Z
  file_id: '13193'
  file_name: matlab-submission.zip
  file_size: 25790
  relation: supplementary_material
  title: Matlab Source Code with Example
file_date_updated: 2023-07-04T08:11:28Z
has_accepted_license: '1'
intvolume: '        42'
isi: 1
issue: '5'
keyword:
- Computer Graphics
- Computational Design
- Computational Geometry
- Shape Modeling
language:
- iso: eng
month: '09'
oa: 1
oa_version: Submitted Version
project:
- _id: 24F9549A-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '715767'
  name: 'MATERIALIZABLE: Intelligent fabrication-oriented Computational Design and
    Modeling'
publication: ACM Transactions on Graphics
publication_identifier:
  eissn:
  - 1557-7368
  issn:
  - 0730-0301
publication_status: published
publisher: Association for Computing Machinery
quality_controlled: '1'
related_material:
  record:
  - id: '12897'
    relation: part_of_dissertation
    status: public
scopus_import: '1'
status: public
title: The design space of Kirchhoff rods
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 42
year: '2023'
...
---
_id: '9817'
abstract:
- lang: eng
  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.
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"
article_number: '126'
article_processing_charge: No
article_type: original
author:
- first_name: Christian
  full_name: Hafner, Christian
  id: 400429CC-F248-11E8-B48F-1D18A9856A87
  last_name: Hafner
- first_name: Bernd
  full_name: Bickel, Bernd
  id: 49876194-F248-11E8-B48F-1D18A9856A87
  last_name: Bickel
  orcid: 0000-0001-6511-9385
citation:
  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>
  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>'
  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>.
  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.
  ista: Hafner C, Bickel B. 2021. The design space of plane elastic curves. ACM Transactions
    on Graphics. 40(4), 126.
  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>.
  short: C. Hafner, B. Bickel, ACM Transactions on Graphics 40 (2021).
conference:
  end_date: 2021-08-13
  location: Virtual
  name: 'SIGGRAF: Special Interest Group on Computer Graphics and Interactive Techniques'
  start_date: 2021-08-09
date_created: 2021-08-08T22:01:26Z
date_published: 2021-07-19T00:00:00Z
date_updated: 2026-04-23T22:30:03Z
day: '19'
ddc:
- '516'
department:
- _id: BeBi
doi: 10.1145/3450626.3459800
ec_funded: 1
external_id:
  isi:
  - '000674930900091'
file:
- access_level: open_access
  checksum: 7e5d08ce46b0451b3102eacd3d00f85f
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  creator: chafner
  date_created: 2021-10-18T10:42:15Z
  date_updated: 2021-10-18T10:42:15Z
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  file_size: 17064290
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  date_created: 2021-10-18T10:42:22Z
  date_updated: 2021-10-18T10:42:22Z
  file_id: '10151'
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file_date_updated: 2021-10-18T10:42:22Z
has_accepted_license: '1'
intvolume: '        40'
isi: 1
issue: '4'
keyword:
- Computing methodologies
- shape modeling
- modeling and simulation
- theory of computation
- computational geometry
- mathematics of computing
- mathematical optimization
language:
- iso: eng
month: '07'
oa: 1
oa_version: Published Version
project:
- _id: 24F9549A-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '715767'
  name: 'MATERIALIZABLE: Intelligent fabrication-oriented Computational Design and
    Modeling'
publication: ACM Transactions on Graphics
publication_identifier:
  eissn:
  - 1557-7368
  issn:
  - 0730-0301
publication_status: published
publisher: Association for Computing Machinery
quality_controlled: '1'
related_material:
  link:
  - description: News on IST Website
    relation: press_release
    url: https://ist.ac.at/en/news/designing-with-elastic-structures/
  record:
  - id: '12897'
    relation: dissertation_contains
    status: public
scopus_import: '1'
status: public
title: The design space of plane elastic curves
tmp:
  image: /images/cc_by.png
  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 40
year: '2021'
...
---
_id: '4097'
abstract:
- lang: eng
  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.
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.
alternative_title:
- LNCS
article_processing_charge: No
author:
- first_name: Herbert
  full_name: Edelsbrunner, Herbert
  id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
  last_name: Edelsbrunner
  orcid: 0000-0002-9823-6833
- first_name: Leonidas
  full_name: Guibas, Leonidas
  last_name: Guibas
- first_name: János
  full_name: Pach, János
  last_name: Pach
- first_name: Richard
  full_name: Pollack, Richard
  last_name: Pollack
- first_name: Raimund
  full_name: Seidel, Raimund
  last_name: Seidel
- first_name: Micha
  full_name: Sharir, Micha
  last_name: Sharir
citation:
  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>'
  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>'
  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.
  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.'
  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.
conference:
  end_date: 1988-07-15
  location: Tampere, Finland
  name: 'ICALP: Automata, Languages and Programming'
  start_date: 1988-07-11
date_created: 2018-12-11T12:06:55Z
date_published: 1988-01-01T00:00:00Z
date_updated: 2022-02-08T10:15:09Z
day: '01'
doi: 10.1007/3-540-19488-6_118
extern: '1'
intvolume: '       317'
keyword:
- line segment
- computational geometry
- Jordan curve
- cell decomposition
- vertical tangency
language:
- iso: eng
main_file_link:
- url: https://link.springer.com/chapter/10.1007/3-540-19488-6_118
month: '01'
oa_version: None
page: 214 - 229
publication: 15th International Colloquium on Automata, Languages and Programming
publication_identifier:
  isbn:
  - 978-3-540-19488-0
publication_status: published
publisher: Springer
publist_id: '2028'
quality_controlled: '1'
scopus_import: '1'
status: public
title: Arrangements of curves in the plane - topology, combinatorics, and algorithms
type: conference
user_id: ea97e931-d5af-11eb-85d4-e6957dddbf17
volume: 317
year: '1988'
...