Discrete Geometry for Reliable Surface Quad-Remeshing

  • Konrad PolthierEmail author
  • Faniry RazafindrazakaEmail author
Conference paper
Part of the Mathematics for Industry book series (MFI, volume 11)


In this overview paper we will glimpse how new concepts from discrete differential geometry help to provide a unifying vertical path through parts of the geometry processing pipeline towards a more reliable interaction. As an example, we will introduce some concepts from discrete differential geometry and the QuadCover algorithm for quadrilateral surface parametrization. QuadCover uses exact discrete differential geometric concepts to convert a pair (simplicial surface, guiding frame field) to a global quad-parametrization of the unstructured surface mesh. Reliability and robustness is an omnipresent issue in geometry processing and computer aided geometric design since its beginning. For example, the variety of incompatible data structures for geometric shapes severely limits a reliable exchange of geometric shapes among different CAD systems as well as a unifying mathematical theory. Here the integrable nature of the discrete differential geometric theory and its presented application to an effective remeshing algorithm may serve an example to envision an increased reliability along the geometry processing pipeline through a consistent processing theory.


Geometry processing Simplicial surfaces Surface parametrization Branched covering QuadCover algorithm Hodge decomposition Minimal surfaces 


  1. 1.
    Bommes, D., Zimmer, H., Kobbelt, L.: Mixed-integer quadrangulation. ACM Trans. Graph. 28(3), 77:1–77:10 (2009)CrossRefGoogle Scholar
  2. 2.
    Bommes, D., Campen, M., Ebke, H.-C., Alliez, P., Kobbelt, L.: Integer-grid maps for reliable quad meshing. ACM Trans. Graph. 32(4), 98:1–98:12 (2013)CrossRefzbMATHGoogle Scholar
  3. 3.
    Campen, M., Kobbelt, L.: Quad layout embedding via aligned parameterization. Comput. Graph. Forum 33, 69–81 (2014)CrossRefGoogle Scholar
  4. 4.
    Hildebrandt, K., Polthier, K.: Generalized shape operators on polyhedral surfaces. Comput. Aided Geom. Design 28(5), 321–343 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Kälberer, F., Nieser, M., Polthier, K.: Quadcover—surface parameterization using branched coverings. Comput. Graph. Forum 26(3), 375–384 (2007)CrossRefzbMATHGoogle Scholar
  6. 6.
    Kälberer, F., Nieser, M., Polthier, K.: Stripe parameterization of tubular surfaces. In: Pascucci, V., Hagen, H., Tierny, J., Tricoche, X. (eds). Topological Methods in Data Analysis and Visualization. Theory, Algorithms, and Applications. Mathematics and Visualization. Springer, New York (2010)Google Scholar
  7. 7.
    Knöppel, F., Crane, K., Pinkall, U., Schröder, P.: Globally optimal direction fields. ACM Trans. Graph. 32(4) (2013)Google Scholar
  8. 8.
    Myles, A., Pietroni, N., Kovacs, D., Zorin, D.: Feature-aligned t-meshes. ACM Trans. Graph. 29(4), 1–11 (2010)CrossRefGoogle Scholar
  9. 9.
    Nieser, M., Reitebuch, U., Polthier, K.: CubeCover—parameterization of 3d volumes. Comput. Graph. Forum 30(5), 1397–1406 (2011)CrossRefGoogle Scholar
  10. 10.
    Nieser, M., Palacios, J., Polthier, K., Zhang, E.: Hexagonal global parameterization of arbitrary surfaces. IEEE Trans. Visual. Comput. Graph. 18(6), 865–878 (2012)CrossRefGoogle Scholar
  11. 11.
    Panozzo, D., Puppo, E., Tarini, M., Sorkine-Hornung, O.: Frame fields: anisotropic and non-orthogonal cross fields. ACM Trans. Graph 33(4), 134:1–134:11 (2014)CrossRefGoogle Scholar
  12. 12.
    Polthier, K.: Unstable periodic discrete minimal surfaces. In: Hildebrandt, S., Karcher, H. (eds). Geometric Analysis and Nonlinear Partial Differential Equations, pp. 127–143. Springer, New York (2002)Google Scholar
  13. 13.
    Polthier, K., Preuss, E.: Identifying vector field singularities using a discrete Hodge decomposition. In: Hege, H.-C., Polthier, K. (eds). Visualization and Mathematics III, pp. 113–134. Springer, New York (2003)Google Scholar

Copyright information

© Springer Japan 2016

Authors and Affiliations

  1. 1.Freie Universität BerlinBerlinGermany

Personalised recommendations