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The Visual Computer

, Volume 34, Issue 6–8, pp 985–995 | Cite as

Wire cut of double-sided minimal surfaces

  • Hao HuaEmail author
  • Tingli Jia
Original Article
  • 128 Downloads

Abstract

We present a systematic method for producing double-sided minimal surfaces by wire-cut machines. A link between minimal surfaces and ruled surfaces is pursued through wire cutting. Weierstrass parameterization is employed to define minimal surfaces (\(\mathbb {R}^3\)) over a complex plane (\(\mathbb {C}\)). Our method consists of three components. First, the orthogonal double-sided cuts match a pair of orthonormal tangent vectors on the surface. Second, A closed-form expression for the principal directions facilitates the global quadrangulation of minimal surfaces. Third, the CNC machine’s toolpath results from the surface’s analytic characterization. Asymptotic cutting and principal cutting are compared in terms of collisions and cutting error. We employed a general-purpose language (Java) to create machine instructions from the Weierstrass representation of minimal surfaces. Thus, the entire workflow from mathematical modeling to production involves no 3D models or CAD/CAM software. Both a 5-axis wire cutter and a customized robotic system were tested.

Keywords

Minimal surface Weierstrass parameterization Wire cut Quadrangulation CNC 

Notes

Acknowledgements

The digital fabrication was carried out at Prof. Li Biao’s Institute of Algorithms & Applications, Southeast University. We would like to thank Wang Xiao for video recoding and the graduate students from the Institute for their support.

Funding

This research is supported by National Natural Science Foundation of China (51778118, 51478116, 51538006, 51578123) and by Ministry of Housing and Urban-Rural Development of China (UDC2017020212).

Compliance with ethical standards

Conflict of interest

The authors declare no potential conflicts of interest.

Human and animal rights statement

This research does not involve human participants or animals.

References

  1. 1.
    Abbena, E., Salamon, S., Gray, A.: Modern Differential Geometry of Curves and Surfaces with Mathematica. CRC Press, Boca Raton (2006)zbMATHGoogle Scholar
  2. 2.
    Alliez, P., Cohen-Steiner, D., Devillers, O., Lévy, B., Desbrun, M.: Anisotropic polygonal remeshing. ACM Trans. Gr. 22(3), 485–493 (2003)CrossRefGoogle Scholar
  3. 3.
    Andersson, S., Hyde, S.T., Larsson, K., Lidin, S.: Minimal surfaces and structures: from inorganic and metal crystals to cell membranes and biopolymers. Chem. Rev. 88(1), 221–242 (1988)CrossRefGoogle Scholar
  4. 4.
    Bo, P., Bartoň, M., Plakhotnik, D., Pottmann, H.: Towards efficient 5-axis flank CNC machining of free-form surfaces via fitting envelopes of surfaces of revolution. Comput. Aided Des. 79, 1–11 (2016)CrossRefGoogle Scholar
  5. 5.
    Brander, D., Bærentzen, J.A., Clausen, K., Fisker, A.S., Gravesen, J., Lund, M.N., Nørbjerg, T.B., Steenstrup, K.H., Søndergaard, A.: Designing for hot-blade cutting: geometric approaches for high-speed manufacturing of doubly-curved architectural surfaces. In: Advances in Architectural Geometry (AAG 2016), pp. 306–327. vdf Hochschulverlag AG an der ETH Zürich (2016)Google Scholar
  6. 6.
    Carberry, E., Fung, K., Glasser, D., Nagle, M., Ordulu, N.: Lecture Notes on Minimal Surfaces (2005). https://pdfs.semanticscholar.org/2ccd/e4f499ea8eba252e173fe14f39458677a4a5.pdf. Accessed 8 May 2018
  7. 7.
    Chen, W., Cai, Y., Zheng, J.: Constructing triangular meshes of minimal area. Comput. Aided Des. Appl. 5(1–4), 508–518 (2008)CrossRefGoogle Scholar
  8. 8.
    Crane, K., Desbrun, M., Schröder, P.: Trivial connections on discrete surfaces. Comput. Gr. Forum 29(5), 1525–1533 (2010)CrossRefGoogle Scholar
  9. 9.
    Dierkes, U., Hildebrandt, S., Sauvigny, F.: Representation formulas and examples of minimal surfaces. In: Dierkes, U.,Hildebrandt, S., Sauvigny, F. (eds.) Minimal Surfaces, pp. 91–236. Springer (2010)Google Scholar
  10. 10.
    Eigensatz, M., Kilian, M., Schiftner, A., Mitra, N.J., Pottmann, H., Pauly, M.: Paneling architectural freeform surfaces. ACM Trans. Gr. 29(4), 45:1–45:10 (2010)CrossRefGoogle Scholar
  11. 11.
    Evans, M.E., Robins, V., Hyde, S.T.: Periodic entanglement I: networks from hyperbolic reticulations. Acta Crystallogr. Sect. A 69(3), 241–261 (2013)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Feringa, J., Søndergaard, A.: Fabricating architectural volume: stereotomic investigations in robotic craft. In:  Gramazio, F., Kohler, M., Langenberg, S. (eds.) Fabricate: Negotiating Design & Making. FABRICATE Conference 2014 at ETH Zürich, pp. 77–83 (2014)Google Scholar
  13. 13.
    Flöry, S., Pottmann, H.: Ruled surfaces for rationalization and design in architecture. In: Sprecher, A., Yeshayahu, S., Lorenzo-Eiroa, P. (eds.) LIFE in: formation. On responsive information and variations in architecture, pp. 103–109 (2010)Google Scholar
  14. 14.
    Fogden, A., Hyde, S.T.: Continuous transformations of cubic minimal surfaces. Eur. Phys. J. B Condens. Matter Complex Syst. 7(1), 91–104 (1999)CrossRefGoogle Scholar
  15. 15.
    Gandy, P.J., Klinowski, J.: Exact computation of the triply periodic G (‘Gyroid’) minimal surface. Chem. Phys. Lett. 321(5–6), 363–371 (2000)CrossRefGoogle Scholar
  16. 16.
    Gandy, P.J., Klinowski, J.: Exact computation of the triply periodic schwarz P minimal surface. Chem. Phys. Lett. 322(6), 579–586 (2000)CrossRefGoogle Scholar
  17. 17.
    Graig, J.J.: Introduction to Robotics: Mechanics and Control. Pearson Prentice Hall, Upper Saddle River (2005)Google Scholar
  18. 18.
    Gramazio, F., Kohler, M., Langenberg, S.: Fabricate: Negotiating Design & making. gta Verlag, Zurich (2014)Google Scholar
  19. 19.
    Harik, R.F., Gong, H., Bernard, A.: 5-Axis flank milling: a state-of-the-art review. Comput. Aided Des. 45(3), 796–808 (2013)CrossRefGoogle Scholar
  20. 20.
    Hua, H.: Javakuka open source library. http://javakuka.org (2016). Accessed 30 Nov 2016
  21. 21.
    Hua, H.: Robotic wire cut. https://youtu.be/7Rh1S-Bs5bw (2017). Accessed 08 June 2017
  22. 22.
    Hyde, S.T.: Crystals: animal, vegetable or mineral? Interface Focus 5(4) (2015). https://doi.org/10.1098/rsfs.2015.0027
  23. 23.
    Knöppel, F., Crane, K., Pinkall, U., Schröder, P.: Globally optimal direction fields. ACM Trans. Gr. 32(4), 59:1–59:10 (2013)CrossRefzbMATHGoogle Scholar
  24. 24.
    Kreyszig, E.: Differential Geometry. Dover Publications, New York (1991)zbMATHGoogle Scholar
  25. 25.
    Lawden, D.F.: Elliptic Functions and Applications, vol. 80. Springer, Berlin (2013)zbMATHGoogle Scholar
  26. 26.
    McGee, W., Feringa, J., Søndergaard, A.: Processes for an architecture of volume: robotic wire cutting. In: Brell-Cokcan, S., Braumann, J. (eds.) Rob | Arch 2012, pp. 62–71. Springer, Association for Robots in Architecture (2012)Google Scholar
  27. 27.
    Meeks III, W., Pérez, J.: The classical theory of minimal surfaces. Bull. Am. Math. Soc. 48(3), 325–407 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Nitsche, J.C.: Lectures on Minimal Surfaces, vol. 1. Cambridge university press, Cambridge (1989)zbMATHGoogle Scholar
  29. 29.
    Osserman, R.: A Survey of Minimal Surfaces. Courier Corporation, New York (2002)zbMATHGoogle Scholar
  30. 30.
    Otto, F., Rasch, B.: Finding Form: Towards an Architecture of the Minimal. Axel Menges, London (1996)Google Scholar
  31. 31.
    Pottmann, H., Schiftner, A., Bo, P., Schmiedhofer, H., Wang, W., Baldassini, N., Wallner, J.: Freeform surfaces from single curved panels. ACM Trans. Gr. 27(3), 76:1–76:10 (2008)CrossRefGoogle Scholar
  32. 32.
    Schoen, A.H.: Infinite periodic minimal surfaces without self-intersections. NASA technical note, NASA-TN-D-5541 (1970)Google Scholar
  33. 33.
    Sharma, R.: The weierstrass representation always gives a minimal surface. arXiv preprint arXiv:1208.5689 (2012)
  34. 34.
    Steenstrup, K.H., Nørbjerg, T.B., Søndergaard, A., Bærentzen, A., Gravesen, J.: Cuttable ruledsurface strips for milling. In: Adriaenssens, S., Gramazio, F., Kohler, M., Menges, A., Pauly, M. (eds.) Advances in Architectural Geometry 2016, pp. 328–342. NCCR Digital Fabrication, Zurich (2016)Google Scholar
  35. 35.
    Taubin, G.: A signal processing approach to fair surface design. In: Mair, S., Cook, R. (eds.) Proceedings of the 22nd Annual Conference on Computer Graphics and Interactive Techniques, pp. 351–358. ACM (1995)Google Scholar
  36. 36.
    Terrones, H.: Computation of minimal surfaces. Le Journal de Physique Colloques 51(C7), C7–345 (1990)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Key Laboratory of Urban and Architectural Heritage Conservation (Southeast University), Ministry of EducationNanjingChina
  2. 2.School of ArchitectureSoutheast UniversityNanjingChina

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