On offsets and curvatures for discrete and semidiscrete surfaces

  • Oleg Karpenkov
  • Johannes Wallner
Original Paper


This paper studies semidiscrete surfaces from the viewpoint of parallelity, offsets, and curvatures. We show how various relevant classes of surfaces are defined by means of an appropriate notion of infinitesimal quadrilateral, how offset surfaces behave in the semidiscrete case, and how to extend and apply the mixed-area based curvature theory which has been developed for polyhedral surfaces.


Semidiscrete surfaces Curvatures 

Mathematics Subject Classification (2000)

53A05 53A10 



The authors are grateful to the anonymous reviewer for many useful suggestions. This research was supported by Austrian Science Fund (grant S9209, part of the National Research Network Industrial Geometry, and grant I705, part of the SFB-Transregio Discretization in Geometry and Dynamics).


  1. Bobenko, A., Pottmann, H., Wallner, J.: A curvature theory for discrete surfaces based on mesh parallelity. Math. Annalen 348, 1–24 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  2. Bobenko, A., Suris, Yu.: Discrete differential geometry: integrable structure, vol. 98. In: Graduate studies in mathematics. American Mathematical Society, Providence (2008)Google Scholar
  3. Cohen-Steiner, D., Morvan, J.M.: Restricted Delaunay triangulations and normal cycle. In: Proceedings of the ACM Symposium on Computational Geometry, pp. 312–321. ACM, New York (2003)Google Scholar
  4. Konopelchenko, B.G., Schief, W.: Three-dimensional integrable lattices in Euclidean spaces: conjugacy and orthogonality. Proc. R. Soc. London A 454, 3075–3104 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 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)CrossRefGoogle Scholar
  6. Müller, C.: Discretizations of the hyperbolic cosine. Beitr. Alg. Geometrie (2013). doi: 10.1007/s13366-012-0097-3
  7. Müller, C., Wallner, J.: Semidiscrete isothermic surfaces. Results Math. (2013). doi: 10.1007/s00025-012-0292-4.
  8. Pottmann, H., Liu, Y., Wallner, J., Bobenko, A., Wang, W.: Geometry of multi-layer freeform structures for architecture. ACM Trans. Graphics 26(3), #65,1–11 (2007)Google Scholar
  9. Pottmann, H., Schiftner, A., Bo, P., Schmiedhofer, H., Wang, W., Baldassini, N., Wallner, J.: Freeform surfaces from single curved panels. ACM Trans. Graphics 27(3), #76,1–10 (2008)Google Scholar
  10. Pottmann, H., Wallner, J.: The focal geometry of circular and conical meshes. Adv. Comp. Math 29, 249–268 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  11. Schief, W.K.: On the unification of classical and novel integrable surfaces. II. Difference geometry. R. Soc. Lond. Proc. Ser. A 459, 373–391 (2003)Google Scholar
  12. Schief, W.K.: On a maximum principle for minimal surfaces and their integrable discrete counterparts. J. Geom. Physics 56, 1484–1495 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  13. Schneider, R.: Convex bodies: the Brunn-Minkowski theory, encyclopedia of mathematics and its applications, vol. 44. Cambridge University Press, Cambridge (1993)Google Scholar
  14. Simon, U., Schwenck-Schellschmidt, A., Viesel, H.: Introduction to the affine differential geometry of hypersurfaces. Lecture Notes. Science University Tokyo (1992)Google Scholar
  15. Wallner, J.: Semidiscrete surface representations. In: Bobenko, A., et al. (eds.) Discrete Differential Geometry, Oberwolfach Reports. Abstracts from the workshop held January 12–17, 2009Google Scholar
  16. Wallner, J.: On the semidiscrete differential geometry of A-surfaces and K-surfaces. J. Geometry 103, 161–176 (2012)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© The Managing Editors 2013

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of LiverpoolLiverpoolUK
  2. 2.Institut für GeometrieTU GrazGrazAustria

Personalised recommendations