Geometriae Dedicata

, Volume 159, Issue 1, pp 207–237 | Cite as

Curvature line parametrized surfaces and orthogonal coordinate systems: discretization with Dupin cyclides

  • Alexander I. Bobenko
  • Emanuel Huhnen-VenedeyEmail author
Original Paper


Cyclidic nets are introduced as discrete analogs of curvature line parametrized surfaces and orthogonal coordinate systems. A 2-dimensional cyclidic net is a piecewise smooth C 1-surface built from surface patches of Dupin cyclides, each patch being bounded by curvature lines of the supporting cyclide. An explicit description of cyclidic nets is given and their relation to the established discretizations of curvature line parametrized surfaces as circular, conical and principal contact element nets is explained. We introduce 3-dimensional cyclidic nets as discrete analogs of triply-orthogonal coordinate systems and investigate them in detail. Our considerations are based on the Lie geometric description of Dupin cyclides. Explicit formulas are derived and implemented in a computer program.


Discrete differential geometry Curvature line parametrized surfaces Orthogonal coordinate systems Dupin cyclides Lie geometry Cyclidic nets Circular nets Principal contact element nets Conical nets 

Mathematics Subject Classification (2000)

51B10 51B25 53A30 37K25 52C26 


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  1. 1.
    Akhmetshin A.A., Vol’vovskij Y.S., Krichever I.M.: Discrete analogs of the Darboux-Egorov metrics. Proc. Steklov Inst. Math. 225, 16–39 (1999)Google Scholar
  2. 2.
    Bauer U., Polthier K., Wardetzky M.: Uniform convergence of discrete curvatures from nets of curvature lines. Discret. Comput. Geom. 43(4), 798–823 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Brannan D.A., Esplen M.F., Gray J.J.: Geometry. Cambridge University Press, Cambridge (2002)Google Scholar
  4. 4.
    Bobenko A.I., Hertrich-Jeromin U.: Orthogonal nets and Clifford algebras. Tohoku Math. Publ. 20, 7–22 (2001)MathSciNetGoogle Scholar
  5. 5.
    Blaschke W.: Vorlesungen über Differentialgeometrie III: Differentialgeometrie der Kreise und Kugeln. Springer, Berlin (1929)zbMATHGoogle Scholar
  6. 6.
    Bobenko A.I., Matthes D., Suris Y.B.: Discrete and smooth orthogonal systems: C -approximation. Int. Math. Res. Not. 45, 2415–2459 (2003)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bobenko, A.I.: Discrete conformal maps and surfaces. In: Clarkson, P.A., Nijhoff F.W. (Eds) Symmetries and integrability of difference equations (Canterbury 1996). London Mathematical Society in Lecture Notes, vol. 255. Cambridge University Press, Cambridge, pp. 97–108 (1999)Google Scholar
  8. 8.
    Bobenko A.I., Suris Yu.B.: On organizing principles of discrete differential geometry. Geometry of spheres. Russ. Math. Surv. 62(1), 1–43 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Bobenko, A.I., Suris, Y.B.: Discrete Differential Geometry. Integrable structure. Graduate Studies in Mathematics, vol. 98. AMS (2008)Google Scholar
  10. 10.
    Cieslinski J., Doliwa A., Santini P.M.: The integrable discrete analogues of orthogonal coordinate systems are multi-dimensional circular lattices. Phys. Lett. A 235, 480–488 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Cecil T.E.: Lie Sphere Geometry. Springer, Berlin (1992)zbMATHGoogle Scholar
  12. 12.
    Degen, W.: Cyclides, Handbook of computer aided geometric design. In: Farin, G., Hoschek, J., Kim, M.-S. (Eds) Elsevier, Amsterdam, xxviii, pp. 575–601 (2002)Google Scholar
  13. 13.
    Dutta D., Martin R.R., Pratt M.J.: Cyclides in surface and solid modeling. IEEE Comput. Graph. Appl. 13, 53–59 (1993)CrossRefGoogle Scholar
  14. 14.
    Doliwa A., Manakov S.V., Santini P.M.: \({\bar\partial}\) -reductions of the multidimensional quadrilateral lattice. The multidimensional circular lattice. Commun. Math. Phys. 196, 1–18 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    jReality Group.: jReality: a Java 3D Viewer for Mathematics, Java class library,
  16. 16.
    Klein, F.: Vorlesungen über höhere Geometrie. 3. Aufl., bearbeitet und herausgegeben von W. Blaschke., VIII+405 S. Berlin, J. Springer (Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen Bd. 22) (German) (1926)Google Scholar
  17. 17.
    Konopelchenko B.G., Schief W.K.: Three-dimensional integrable lattices in Euclidean spaces: conjugacy and orthogonality. R. Soc. Lond. Proc. Ser. A 454, 3075–3104 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Lávička M., Vršek J.: On the representation of Dupin cyclides in Lie sphere geometry with applications. J. Geom. Graph. 13(2), 145–162 (2009)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Liu Y., Pottmann H., Wallner J., Yang Y.-L., Wang W.: Geometric modeling with conical meshes and developable surfaces. Proc. SIGGRAPH 25, 681–689 (2006)CrossRefGoogle Scholar
  20. 20.
    Martin, R.R.: Principal patches—a new class of surface patch based on differential geometry. Eurograph. Proc. (1983)Google Scholar
  21. 21.
    McLean, D.: A method of generating surfaces as a composite of cyclide patches. Comput. J. 28(4) (1985)Google Scholar
  22. 22.
    Martin, R.R., de Pont, J., Sharrock, T.J.: Cyclide surfaces in computer aided design. The mathematics of surfaces. Clarendon Press, Oxford, pp. 253–267 (1986)Google Scholar
  23. 23.
    Nutbourne A.W., Martin R.R.: Differential Geometry Applied to Curve and Surface Design. Horwood, Chichester (1988)Google Scholar
  24. 24.
    Pinkall U.: Dupin hypersurfaces. Math. Ann. 270(3), 427–440 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Pinkall, U.: Dupinsche Zykliden. In: Fischer, G. (ed) Mathematische Modelle. Vieweg, pp. 30–32 (1986)Google Scholar
  26. 26.
    Pottmann, H., Wallner, J.: The focal geometry of circular and conical meshes. Adv. Comp. Math (2007, to appear)Google Scholar
  27. 27.
    Srinivas Y.L., Kumar V., Dutta D.: Surface design using cyclide patches. Comput. Aided Des. 28(4), 263–276 (1996)CrossRefGoogle Scholar
  28. 28.
    TU-Berlin: Java Tools for Experimental Mathematics,
  29. 29.
    Zakharov V.E.: Description of the n-orthogonal curvilinear coordinate systems and Hamiltonian integrable systems of hydrodynamic type. I. Integration of the Lamé equations. Duke Math. J. 94(1), 103–139 (1998)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Institut für Mathematik, MA 8-3Technische Universität BerlinBerlinGermany

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