Skip to main content
SpringerLink
Log in

Discrete Surfaces in Isotropic Geometry

  • Conference paper
Mathematics of Surfaces XII (Mathematics of Surfaces 2007)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4647))

Included in the following conference series:

Abstract

Meshes with planar quadrilateral faces are desirable discrete surface representations for architecture. The present paper introduces new classes of planar quad meshes, which discretize principal curvature lines of surfaces in so-called isotropic 3-space. Like their Euclidean counterparts, these isotropic principal meshes meshes are visually expressing fundamental shape characteristics and they can satisfy the aesthetical requirements in architecture. The close relation between isotropic geometry and Euclidean Laguerre geometry provides a link between the new types of meshes and the known classes of conical meshes and edge offset meshes. The latter discretize Euclidean principal curvature lines and have recently been realized as particularly suited for freeform structures in architecture, since they allow for a supporting beam layout with optimal node properties. We also present a discrete isotropic curvature theory which applies to all types of meshes including triangle meshes. The results are illustrated by discrete isotropic minimal surfaces and meshes computed by a combination of optimization and subdivision.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Blaschke, W.: Vorlesungen über Differentialgeometrie, vol. 3. Springer, Heidelberg (1929)

    MATH  Google Scholar 

  2. Bobenko, A., Hoffmann, T., Springborn, B.: Minimal surfaces from circle patterns: Geometry from combinatorics. Ann. of Math. 164, 231–264 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bobenko, A., Pinkall, U.: Discrete isothermic surfaces. J. Reine Angew. Math. 475, 187–208 (1996)

    MATH  MathSciNet  Google Scholar 

  4. Bobenko, A., Suris, Y.: Discrete differential geometry. Consistency as integrability. (2005), monograph pre-published at http://arxiv.org/abs/math.DG/0504358

  5. Bobenko, A., Suris, Y.: On organizing principles of discrete differential geometry, geometry of spheres (to appear)

    Google Scholar 

  6. Bobenko, A., Springborn, B.: Variational principles for circle patterns and Koebe’s theorem. Trans. Amer. Math. Soc. 356, 659–689 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  7. Brell-Cokcan, S., Pottmann, H.: Tragstruktur für Freiformflächen in Bauwerken (Patent No. A1049/2006) (2006)

    Google Scholar 

  8. Cecil, T.: Lie Sphere Geometry. Springer, Heidelberg (1992)

    MATH  Google Scholar 

  9. Cohen-Steiner, D., Alliez, P., Desbrun, M.: Variational shape approximation. ACM Trans. Graphics 23(3), 905–914 (2004)

    Article  Google Scholar 

  10. Cutler, B., Whiting, E.: Constrained planar remeshing for architecture. In: Symp. Geom. Processing. poster (2006)

    Google Scholar 

  11. Glymph, J., et al.: A parametric strategy for freeform glass structures using quadrilateral planar facets. In: Glymph, J. (ed.) Acadia 2002, pp. 303–321. ACM Press, New York (2002)

    Google Scholar 

  12. Koenderink, I.J., van Doorn, A.J.: Image processing done right. In: Heyden, A., Sparr, G., Nielsen, M., Johansen, P. (eds.) ECCV 2002. LNCS, vol. 2350, pp. 158–172. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

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

    Article  Google Scholar 

  14. Martin, R., de Pont, J., Sharrock, T.: Cyclide surfaces in computer aided design. In: Gregory, J.A. (ed.) The mathematics of surfaces, pp. 253–268. Clarendon Press, Oxford (1986)

    Google Scholar 

  15. Peternell, M.: Developable surface fitting to point clouds. Comp. Aid. Geom. Des. 21(8), 785–803 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  16. Peternell, M., Pottmann, H.: A Laguerre geometric approach to rational offsets. Comp. Aid. Geom. Des. 15, 223–249 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  17. Pottmann, H., Opitz, K.: Curvature analysis and visualization for functions defined on Euclidean spaces or surfaces. Comp. Aid. Geom. Des. 11, 655–674 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  18. Pottmann, H., Peternell, M.: Applications of Laguerre geometry in CAGD. Comp. Aid. Geom. Des. 15, 165–186 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  19. Pottmann, H., Brell-Cokcan, S., Wallner, J.: Discrete surfaces for architectural design. In: Curve and Surface Design: Avignon 2006, pp. 213–234. Nashboro Press (2006)

    Google Scholar 

  20. Pottmann, H., Wallner, J.: The focal geometry of circular and conical meshes. Adv. Comput. Math. (to appear)

    Google Scholar 

  21. Pottmann, H., Liu, Y., Wallner, J., Bobenko, A., Wang, W.: Geometry of multi-layer freeform structures for architecture. ACM Trans. Graphics 26(3) (2007)

    Google Scholar 

  22. Sachs, H.: Isotrope Geometrie des Raumes. Vieweg (1990)

    Google Scholar 

  23. Santalo, L.: Integral Geometry and Geometric Probability. Addison-Wesley, London (1976)

    MATH  Google Scholar 

  24. Sauer, R.: Differenzengeometrie. Springer, Heidelberg (1970)

    MATH  Google Scholar 

  25. Schneider, R.: Convex bodies: the Brunn- Minkowski theory. Cambridge University Press, Cambridge (1993)

    MATH  Google Scholar 

  26. Schramm, O.: Circle patterns with the combinatorics of the square grid. Duke Math. J. 86, 347–389 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  27. Schramm, O.: How to cage an egg. Invent. Math. 107, 543–560 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  28. Simon, U., Schwenck-Schellschmidt, A., Viesel, H.: Introduction to the affine differential geometry of hypersurfaces. Lecture Notes. Science Univ. Tokyo (1992)

    Google Scholar 

  29. Strubecker, K.: Differentialgeometrie des isotropen Raumes III: Flächentheorie. Math. Zeitschrift 48, 369–427 (1942)

    Article  MathSciNet  Google Scholar 

  30. Strubecker, K.: Airy’sche Spannungsfunktion und isotrope Differentialgeometrie. Math. Zeitschrift 78, 189–198 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  31. Strubecker, K.: Über das isotrope Gegenstück z = 32 J(x + iy)2/3 der Minimalfläche von Enneper. Abh. Math. Sem. Univ. Hamburg 44, 152–174 (1975/76)

    Google Scholar 

  32. Strubecker, K.: Duale Minimalflächen des isotropen Raumes. Rad JAZU 382, 91–107 (1978)

    MathSciNet  Google Scholar 

  33. Wallner, J., Pottmann, H.: Infinitesimally flexible meshes and discrete minimal surfaces. Monatsh. Math. (to appear)

    Google Scholar 

  34. Ziegler, G.: Lectures on Polytopes. Springer, Heidelberg (1995)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Geometric Modeling and Industrial Geometry, Vienna University of Technology, Wiedner Hauptstr. 8-10, A-1040 Wien, Austria

    Helmut Pottmann

  2. Department of Computer Science, University of Hong Kong, China

    Yang Liu

Authors
  1. Helmut Pottmann
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yang Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Ralph Martin Malcolm Sabin Joab Winkler

Rights and permissions

Reprints and permissions

Copyright information

© 2007 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Pottmann, H., Liu, Y. (2007). Discrete Surfaces in Isotropic Geometry. In: Martin, R., Sabin, M., Winkler, J. (eds) Mathematics of Surfaces XII. Mathematics of Surfaces 2007. Lecture Notes in Computer Science, vol 4647. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73843-5_21

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-73843-5_21

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-73842-8

  • Online ISBN: 978-3-540-73843-5

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation