---
_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:
- access_level: open_access
  checksum: 0dc9f5a6422b8a49a79026900f349ee5
  content_type: application/pdf
  creator: chafner
  date_created: 2024-07-05T12:05:17Z
  date_updated: 2024-07-05T12:05:17Z
  file_id: '17204'
  file_name: sif-final.pdf
  file_size: 7225150
  relation: main_file
  success: 1
- access_level: open_access
  checksum: cde433c6a40688d5f1187fb5721f6f94
  content_type: application/pdf
  creator: chafner
  date_created: 2024-07-05T12:06:03Z
  date_updated: 2024-07-05T12:06:03Z
  file_id: '17205'
  file_name: sif-supp-final.pdf
  file_size: 397262
  relation: supplementary_material
- access_level: open_access
  checksum: c0457a09c2ab9a1c2935c995dcc84907
  content_type: video/mp4
  creator: chafner
  date_created: 2024-07-17T09:29:13Z
  date_updated: 2024-07-17T09:29:13Z
  file_id: '17276'
  file_name: sif-video-final.mp4
  file_size: 170001305
  relation: supplementary_material
  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'
...