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Multi-Nets. Classification of Discrete and Smooth Surfaces with Characteristic Properties on Arbitrary Parameter Rectangles

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Abstract

We investigate the common underlying discrete structures for various smooth and discrete nets. The main idea is to impose the characteristic properties of the nets not only on elementary quadrilaterals but also on arbitrary parameter rectangles. For discrete planar quadrilateral nets, circular nets, \(Q^*\)-nets and conical nets we obtain a characterization of the corresponding discrete multi-nets. In the limit these discrete nets lead to some classical classes of smooth surfaces. Furthermore, we propose to use the characterized discrete nets as discrete extensions for the nets to obtain structure preserving subdivision schemes.

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References

  1. Blaschke, W.: Vorlesungen über Differentialgeometrie III. Grundlehren der Mathematischen Wissenschaften, vol. 29. Springer, Berlin (1929)

    MATH  Google Scholar 

  2. Bo, P., Liu, Y., Tu, C., Zhang, C., Wang, W.: Surface fitting with cyclide splines. Comput. Aided Geom. Des. 43(Suppl C), 2–15 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bo, P., Pottmann, H., Kilian, M., Wang, W., Wallner, J.: Circular arc structures. ACM Trans. Graph. 30(4), 101:1–101:12 (2011)

    Article  Google Scholar 

  4. Bobenko, A.I., Huhnen-Venedey, E.: Curvature line parametrized surfaces and orthogonal coordinate systems: discretization with Dupin cyclides. Geom. Dedicata 159, 207–237 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bobenko, A.I., Huhnen-Venedey, E., Rörig, T.: Supercyclidic nets. Int. Math. Res. Not. IMRN 2017(2), 323–371 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bobenko, A.I., Matthes, D., Suris, Yu.B.: Discrete and smooth orthogonal systems: \(C^\infty \)-approximation. Int. Math. Res. Not. IMRN 2003(45), 2415–2459 (2003)

  7. Bobenko, A.I., Pottmann, H., Wallner, J.: A curvature theory for discrete surfaces based on mesh parallelity. Math. Ann. 348(1), 1–24 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bobenko, A.I., Suris, Yu.B.: Isothermic surfaces in sphere geometries as Moutard nets. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 463(2088), 3171–3193 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bobenko, A.I., Suris, Yu.B.: Discrete Differential Geometry. Graduate Studies in Mathematics, vol. 98. American Mathematical Society, Providence (2008)

  10. Bol, G.: Projektive Differentialgeometrie I, vol. VII. Vandenhoeck & Ruprecht, Göttingen (1951)

    MATH  Google Scholar 

  11. Bouaziz, S., Deuss, M., Schwartzburg, Y., Weise, T., Pauly, M.: Shape-up: shaping discrete geometry with projections. Comput. Graph. Forum 31(5), 1657–1667 (2012)

    Article  Google Scholar 

  12. Degen, W.: Nets with plane silhouettes. In: Fisher, R.B. (ed.) Proceedings of the 5th IMA Conference on the Mathematics of Surfaces. The Institute of Mathematics and its Applications Conference Series No. S., vol. 50, pp. 117–133. Clarendon Press, Oxford (1994)

  13. Degen, W.: Nets with plane silhouettes II. In: Goodman, T., Martin, R. (eds.) The Mathematics of Surfaces VII, pp. 263–279. Information Geometers, Limited, Winchester (1997)

    Google Scholar 

  14. Doliwa, A., Santini, P.M.: Multidimensional quadrilateral lattices are integrable. Phys. Lett. A 233(4–6), 265–372 (1997)

    MathSciNet  MATH  Google Scholar 

  15. Douthe, C., Mesnil, R., Orts, H., Baverel, O.: Isoradial meshes: covering elastic gridshells with planar facets. Autom. Constr. 83(Suppl C), 222–236 (2017)

    Article  Google Scholar 

  16. Dyn, N., Levin, D., Gregory, J.A.: A butterfly subdivision scheme for surface interpolation with tension control. ACM Trans. Graph. 9(2), 160–169 (1990)

    Article  MATH  Google Scholar 

  17. Grohs, P.: Smoothness analysis of subdivision schemes on regular grids by proximity. SIAM J. Numer. Anal. 46(4), 2169–2182 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  18. Hertrich-Jeromin, U.: Introduction to Möbius Differential Geometry. London Mathematical Society Lecture Note Series, vol. 300. Cambridge University Press, Cambridge (2003)

    Book  MATH  Google Scholar 

  19. Huhnen-Venedey, E., Rörig, T.: Discretization of asymptotic line parametrizations using hyperboloid surface patches. Geom. Dedicata 168(1), 265–289 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kobbelt, L.: Interpolatory subdivision on open quadrilateral nets with arbitrary topology. Comput. Graph. Forum 15(3), 409–420 (1996)

    Article  Google Scholar 

  21. Liu, Y., Pottmann, H., Wallner, J., Yang, Y.-L., Wang, W.: Geometric modeling with conical meshes and developable surfaces. ACM Trans. Graph. 25(3), 681–689 (2006)

    Article  Google Scholar 

  22. Mesnil, R., Douthe, C., Baverel, O., Léger, B.: Marionette mesh: from descriptive geometry to fabrication-aware design. In: Adriaenssens, S., et al. (eds.) Advances in Architectural Geometry 2016, pp. 62–81. Hochschulverlag, Zürich (2016)

    Google Scholar 

  23. Sauer, R.: Projektive Liniengeometrie. de Gruyter, Berlin (1937)

    MATH  Google Scholar 

  24. Tang, C., Sun, X., Gomes, A., Wallner, J., Pottmann, H.: Form-finding with polyhedral meshes made simple. ACM Trans. Graph. (2014). https://doi.org/10.1145/2601097.2601213

    Article  MATH  Google Scholar 

  25. Vaxman, A., Müller, C., Weber, O.: Canonical Möbius subdivision. ACM Trans. Graph. (2018). https://doi.org/10.1145/3272127.3275007

    Article  Google Scholar 

  26. Zadravec, M., Schiftner, A., Wallner, J.: Designing quad-dominant meshes with planar faces. Comput. Graph. Forum (2010). https://doi.org/10.2312/CGF.v29i5pp1671-1679

  27. Zorin, D., Schröder, P., Sweldens, W.: Interpolating subdivision for meshes with arbitrary topology. In: Proceedings of the 23rd Annual Conference on Computer Graphics and Interactive Techniques (SIGGRAPH’96), pp. 189–192. ACM, New York (1996)

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Acknowledgements

We would like to thank Wolfgang Schief and Jan Techter for many fruitful discussions. This research was supported by the DFG Collaborative Research Center TRR 109, “Discretization in Geometry and Dynamics.” – www.discretization.de. Helmut Pottmann was supported through Grant I 2978-N35 of the Austrian Science Fund (FWF).

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Authors and Affiliations

  1. Institute of Mathematics, TU Berlin, Secr. MA 8-4, 10623, Berlin, Germany

    Alexander I. Bobenko & Thilo Rörig

  2. Visual Computing Center, King Abdullah University of Science and Technology, Thuwal, 23955-6900, Saudi Arabia

    Helmut Pottmann

  3. Geometric Modeling and Industrial Geometry, TU Wien, 1040, Vienna, Austria

    Helmut Pottmann

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  1. Alexander I. Bobenko
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  2. Helmut Pottmann
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  3. Thilo Rörig
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Correspondence to Thilo Rörig.

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Bobenko, A.I., Pottmann, H. & Rörig, T. Multi-Nets. Classification of Discrete and Smooth Surfaces with Characteristic Properties on Arbitrary Parameter Rectangles. Discrete Comput Geom 63, 624–655 (2020). https://doi.org/10.1007/s00454-019-00101-1

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  • DOI: https://doi.org/10.1007/s00454-019-00101-1

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