research-article

Open access

Freeform quad-based kirigami

Authors:

Helmut Pottmann,

Johannes WallnerAuthors Info & Claims

ACM Transactions on Graphics (TOG), Volume 39, Issue 6

Article No.: 209, Pages 1 - 11

https://doi.org/10.1145/3414685.3417844

Published: 27 November 2020 Publication History

Abstract

Kirigami, the traditional Japanese art of paper cutting and folding generalizes origami and has initiated new research in material science as well as graphics. In this paper we use its capabilities to perform geometric modeling with corrugated surface representations possessing an isometric unfolding into a planar domain after appropriate cuts are made. We initialize our box-based kirigami structures from orthogonal networks of curves, compute a first approximation of their unfolding via mappings between meshes, and complete the process by global optimization. Besides the modeling capabilities we also study the interesting geometry of special kirigami structures from the theoretical side. This experimental paper strives to relate unfoldable checkerboard arrangements of boxes to principal meshes, to the transformation theory of discrete differential geometry, and to a version of the Gauss theorema egregium.

Supplementary Material

MP4 File (a209-jiang.mp4)

Download
161.83 MB

MP4 File (3414685.3417844.mp4)

Presentation video

Download
47.43 MB

References

[1]

Alexander Bobenko and Yuri Suris. 2009. Discrete differential geometry: Integrable Structure. Number 98. American Math. Soc.

[2]

David Bommes, Henrik Zimmer, and Leif Kobbelt. 2009. Mixed-integer Quadrangulation. ACM Trans. Graph. 28, 3 (2009), #77,1--10.

Digital Library

[3]

Mario Botsch, Stephan Steinberg, Stephan Bischoff, and Leif Kobbelt. 2002. OpenMesh: A Generic and Efficient Polygon Mesh Data Structure. Proc. OpenSG Symposium.

[4]

Toen Castle, Yigil Cho, Xingting Gong, Euiyeon Jung, Daniel M. Sussman, Shu Yang, and Randall D. Kamien. 2014. Making the Cut: Lattice Kirigami Rules. Phys. Rev. Lett 113, 245502 (2014), 1--5.

[5]

Toen Castle, Daniel M. Sussman, Michael Tanis, and Randall D. Kamien. 2016. Additive lattice kirigami. Sci. Advances 2, e1601258 (2016), 1--11.

[6]

Frédéric Cazals and Marc Pouget. 2004. Smooth surfaces, umbilics, lines of curvatures, foliations, ridges and the medial axis: a concise overview. Research Report RR-5138. INRIA. https://hal.inria.fr/inria-00071445/

[7]

Gary Choi, Levi Dudte, and L. Mahadevan. 2019. Programming shape using kirigami tessellations. Nature Materials 18 (2019), 999--1004.

[8]

David Cohen-Steiner and Jean Marie Morvan. 2003. Restricted Delaunay triangulations and normal cycle. In Proc. ACM Symp. Comp. Graphics. 312--321.

Digital Library

[9]

Erik D. Demaine and Joseph O'Rourke. 2007. Geometric Folding Algorithms. Cambridge University Press.

Digital Library

[10]

Erik D. Demaine and Tomohiro Tachi. 2017. Origamizer: A Practical Algorithm for Folding Any Polyhedron. In Proc. 33rd SoCG. 1--15.

[11]

Manfredo do Carmo. 1976. Differential Geometry of Curves and Surfaces. Prentice-Hall.

[12]

Levi H. Dudte, Etienne Vouga, Tomohiro Tachi, and L. Mahadevan. 2016. Programming curvature using origami tessellations. Nature Materials 15 (2016), 583--588.

[13]

Ruslan Guseinov, Connor McMahan, Jesús Pérez, Chiara Daraio, and Bernd Bickel. 2020. Programming temporal morphing of self-actuated shells. Nature Communications 11 (2020), #237,1--7.

[14]

Ruslan Guseinov, Eder Miguel, and Bernd Bickel. 2017. CurveUps: shaping objects from flat plates with tension-actuated curvature. ACM Trans. Graph. 36, 4 (2017), #64,1--12.

Digital Library

[15]

Alec Jacobson, Daniele Panozzo, et al. 2018. libigl: A simple C++ geometry processing library. https://libigl.github.io

[16]

Caigui Jiang, Klara Mundilova, Florian Rist, Johannes Wallner, and Helmut Pottmann. 2019a. Curve-pleated structures. ACM Trans. Graph. 38, 6 (2019), #169,1--13.

Digital Library

[17]

Caigui Jiang, Chi-Han Peng, Peter Wonka, and Helmut Pottmann. 2019b. Checkerboard patterns with black rectangles. ACM Trans. Graph. 38, 6 (2019), #171,1--13.

[18]

Caigui Jiang, Cheng Wang, Florian Rist, Johannes Wallner, and Helmut Pottmann. 2020. Quad-mesh based isometric mappings and developable surfaces. ACM Trans. Graph. 39, 4 (2020), #128,1--13.

Digital Library

[19]

Mina Konaković, Keenan Crane, Bailin Deng, Sophien Bouaziz, Daniel Piker, and Mark Pauly. 2016. Beyond Developable: Computational Design and Fabrication with Auxetic Materials. ACM Trans. Graph. 35, 4 (2016).

Digital Library

[20]

Mina Konaković-Luković, Julian Panetta, Keenan Crane, and Mark Pauly. 2018. Rapid Deployment of Curved Surfaces via Programmable Auxetics. ACM Trans. Graph. 37, 4 (2018), #106,1--13.

Digital Library

[21]

Ligang Liu, Lei Zhang, Yin Xu, Craig Gotsman, and Steven Gortler. 2008. A local/global approach to mesh parameterization. Comput. Graph. Forum 27, 5 (2008), 1495--1504.

[22]

Kaj Madsen, Hans Bruun Nielsen, and Ole Tingleff. 2004. Methods for non-linear least squares problems (2nd ed.). Technical Univ. Denmark.

[23]

Tanmoy Mukhopadhyay et al. 2020. Programmable stiffness and shape modulation in origami materials: emergence of a distant actuation feature. Applied Materials Today 19, 100537 (2020), 1--7.

[24]

Helmut Pottmann, Yang Liu, Johannes Wallner, Alexander Bobenko, and Wenping Wang. 2007. Geometry of Multi-layer Freeform Structures for Architecture. ACM Trans. Graph. 26, 3 (2007), #65,1--11.

Digital Library

[25]

Nicolas Ray, Bruno Vallet, Wan Chiu Li, and Bruno Lévy. 2008. N-Symmetry Direction Field Design. ACM Trans. Graph. 27, 2 (2008), #10,1--13.

Digital Library

[26]

Keith A. Seffen. 2012. Compliant shell mechanisms. Phil. Trans. R. Soc. A 370 (2012), 2010--2026.

[27]

Olga Sorkine and Marc Alexa. 2007. As-rigid-as-possible surface modeling. In Proc. Symposium Geometry Processing. 109--116.

[28]

Daniel M. Sussman, Yigil Cho, Toen Castle, Xingting Gong, Euiyeon Jung, Shu Yang, and Randall D. Kamien. 2015. Algorithmic lattice kirigami: A route to pluripotent materials. PNAS 112, 24 (2015), 7449--7453.

[29]

Tomohiro Tachi. 2009. Generalizationof Rigid Foldable Quadrilateral Mesh Origami. In Proc. IASS Symposium. Univ. Politècnica de València, 2287--2294.

[30]

Tomohiro Tachi. 2010a. Freeform Variations of Origami. J. Geom. Graphics 14 (2010), 203--215.

[31]

Tomohiro Tachi. 2010b. Origamizing Polyhedral Surfaces. IEEE Trans. Vis. Comput. Graph. 16 (2010), 298--311.

Digital Library

[32]

Yichao Tang, Yanbin Li, Yaoye Hong, and Jie Yin. 2019. Programmable active kirigami metasheets with more freedom of actuation. PNAS 116 (2019), 26407--13.

[33]

Sivan Toledo. 2003. Taucs, A Library of Sparse Linear Solvers. http://www.tau.ac.il/~stoledo/taucs

[34]

Fei Wang, Xiaogang Guo, Jingxian Xu, Yihui Zhang, and C. Q. Chen. 2017. Patterning curved three-dimensional structures with programmable kirigami designs. J. Appl. Mech. 84, 061007 (2017), 1--7.

[35]

Zhiyuan Wei, Zengcai V. Guo, Levi Dudte, Hiyi Liang, and L. Mahadevan. 2013. Geometric Mechanics of Periodic Pleated Origami. Phys. Rev. Lett. 110, 215501 (2013), 1--5.

[36]

Ruikang Xie, Yan Chen, and Joseph M. Gattas. 2015. Parametrisation and application of cube and eggbox-type folded geometries. Intl. J. Space Structures 30 (2015), 99--110.

[37]

Yihui Zhang et al. 2015. A mechanically driven form of kirigami as a route to 3D mesostructures in micro/nanomembranes. PNAS 112 (2015), 11757--64.

Cited By

Ceballos Inza VFykouras PRist FHäseker DHojjat MMüller CPottmann H(2024)Designing triangle meshes with controlled roughnessACM Transactions on Graphics10.1145/368794043:6(1-20)Online publication date: 19-Dec-2024
https://dl.acm.org/doi/10.1145/3687940
Segall ARen JVaxman ASorkine-Hornung O(2024)Fabric Tessellation: Realizing Freeform Surfaces by SmockingACM Transactions on Graphics10.1145/365815143:4(1-20)Online publication date: 19-Jul-2024
https://doi.org/10.1145/3658151
Ren JSegall ASorkine-Hornung O(2023)Digital 3D Smocking DesignACM Transactions on Graphics10.1145/3631945Online publication date: 16-Nov-2023
https://doi.org/10.1145/3631945
Show More Cited By

Index Terms

Freeform quad-based kirigami
1. Computing methodologies
  1. Computer graphics
    1. Shape modeling
  2. Symbolic and algebraic manipulation
    1. Symbolic and algebraic algorithms
      1. Optimization algorithms

Recommendations

Quad-mesh based isometric mappings and developable surfaces

We discretize isometric mappings between surfaces as correspondences between checkerboard patterns derived from quad meshes. This method captures the degrees of freedom inherent in smooth isometries and enables a natural definition of discrete ...
Freeform surfaces from single curved panels

Motivated by applications in architecture and manufacturing, we discuss the problem of covering a freeform surface by single curved panels. This leads to the new concept of semi-discrete surface representation, which constitutes a link between smooth ...
Geometry of multi-layer freeform structures for architecture
SIGGRAPH '07: ACM SIGGRAPH 2007 papers

The geometric challenges in the architectural design of freeform shapes come mainly from the physical realization of beams and nodes. We approach them via the concept of parallel meshes, and present methods of computation and optimization. We discuss ...

Comments

Information & Contributors

Information

Published In

cover image ACM Transactions on Graphics

ACM Transactions on Graphics Volume 39, Issue 6

December 2020

1605 pages

ISSN:0730-0301

EISSN:1557-7368

DOI:10.1145/3414685

Editor:
Karol Myszkowski
MPI Informatik

Issue’s Table of Contents

Copyright © 2020 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected].

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 27 November 2020

Published in TOG Volume 39, Issue 6

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Research-article

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

17
Total Citations
View Citations
1,398
Total Downloads

Downloads (Last 12 months)390
Downloads (Last 6 weeks)38

Reflects downloads up to 11 Feb 2025

Other Metrics

View Author Metrics

Citations

Cited By

Ceballos Inza VFykouras PRist FHäseker DHojjat MMüller CPottmann H(2024)Designing triangle meshes with controlled roughnessACM Transactions on Graphics10.1145/368794043:6(1-20)Online publication date: 19-Dec-2024
https://dl.acm.org/doi/10.1145/3687940
Segall ARen JVaxman ASorkine-Hornung O(2024)Fabric Tessellation: Realizing Freeform Surfaces by SmockingACM Transactions on Graphics10.1145/365815143:4(1-20)Online publication date: 19-Jul-2024
https://doi.org/10.1145/3658151
Ren JSegall ASorkine-Hornung O(2023)Digital 3D Smocking DesignACM Transactions on Graphics10.1145/3631945Online publication date: 16-Nov-2023
https://doi.org/10.1145/3631945
Keros ASubr K(2023)Spectral Coarsening with Hodge LaplaciansACM SIGGRAPH 2023 Conference Proceedings10.1145/3588432.3591544(1-11)Online publication date: 23-Jul-2023
https://dl.acm.org/doi/10.1145/3588432.3591544
Zhou JKim RDoubrovski ZMartins JGiaccardi EKarana E(2023)Cyano-chromic Interface: Aligning Human-Microbe Temporalities Towards Noticing and Attending to Living ArtefactsProceedings of the 2023 ACM Designing Interactive Systems Conference10.1145/3563657.3596132(820-838)Online publication date: 10-Jul-2023
https://dl.acm.org/doi/10.1145/3563657.3596132
Huang FLiu CHsiao KKuo YChu HYang Y(2023)Image-Based OA-Style Paper Pop-Up Design via Mixed-Integer ProgrammingIEEE Transactions on Visualization and Computer Graphics10.1109/TVCG.2022.318956929:10(4269-4283)Online publication date: 1-Oct-2023
https://doi.org/10.1109/TVCG.2022.3189569
Eguchi SOkabe COhira MTanaka H(2022)Pneumatic Auxetics:Extended Abstracts of the 2022 CHI Conference on Human Factors in Computing Systems10.1145/3491101.3519801(1-7)Online publication date: 27-Apr-2022
https://dl.acm.org/doi/10.1145/3491101.3519801
Dang XFeng FDuan HWang J(2022)Theorem on the Compatibility of Spherical Kirigami TessellationsPhysical Review Letters10.1103/PhysRevLett.128.035501128:3Online publication date: 21-Jan-2022
https://doi.org/10.1103/PhysRevLett.128.035501
McMahan CAkerson ACelli PAudoly BDaraio C(2022)Effective continuum models for the buckling of non-periodic architected sheets that display quasi-mechanism behaviorsJournal of the Mechanics and Physics of Solids10.1016/j.jmps.2022.104934166(104934)Online publication date: Sep-2022
https://doi.org/10.1016/j.jmps.2022.104934
Jiang CRist FWang HWallner JPottmann H(2022)Shape-morphing mechanical metamaterialsComputer-Aided Design10.1016/j.cad.2021.103146143:COnline publication date: 9-Apr-2022
https://dl.acm.org/doi/10.1016/j.cad.2021.103146
Show More Cited By

View Options

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Article

Figures

Tables

Media

View Issue’s Table of Contents