The Visual Computer

, Volume 24, Issue 7–9, pp 745–752 | Cite as

Sketching freeform meshes using graph rotation functions

  • Bin Sheng
  • Enhua Wu
  • Hanqiu Sun
Original Article


We present a new approach for sketching free form meshes with topology consistency. Firstly, we interpret the given 2D curve to be the projection of the 3D curve with the minimum curvature. Then we adopt a topology-consistent strategy based on the graph rotation system, to trace the simple faces on the interconnecting 3D curves. With the face tracing algorithm, our system can identify the 3D surfaces automatically. After obtaining the boundary curves for the faces, we apply Delaunay triangulation on these faces. Finally, the shape of the triangle mesh that follows the 3D boundary curves is computed by using harmonic interpolation. Meanwhile our system provides real-time algorithms for both control curve generation and the subsequent surface optimization. With the incorporation of topological manipulation into geometrical modeling, we show that automatically generated models are both beneficial and feasible.


Freeform surfaces Subdivision Laplace’s equation 


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  1. 1.
    Autodesk: 3DS MAX. (2008)Google Scholar
  2. 2.
    Autodesk: MAYA. (2008)Google Scholar
  3. 3.
    Baumgart, B.G.: Winged edge polyhedron representation. Tech. rep., Stanford, CA, USA (1972)Google Scholar
  4. 4.
    George, P.L., Borouchaki, H.: Delaunay Triangulation and Meshing. Hermes, Paris, France (1998)zbMATHGoogle Scholar
  5. 5.
    Hoffmann, C.M., Vanecek, G.: Fundamental techniques for geometric and solid modeling. Manufact. Autimation Syst.: Techniques Technol. 48, 157–160 (1990)Google Scholar
  6. 6.
    Igarashi, T., Hughes, J.: Smooth meshes for sketch-based freeform modeling. In: Proceedings of the 2003 Symposium on Interactive 3D Graphics, pp. 139–142. ACM, New York, NY (2003)CrossRefGoogle Scholar
  7. 7.
    Igarashi, T., Matsuoka, S., Tanaka, H.: Teddy: A sketching interface for 3D freeform design. In: Proceedings of ACM SIGGRAPH 1999, pp. 409–416. ACM, New York, NY (1999)Google Scholar
  8. 8.
    Karpenko, O.A., Hughes, J.F.: Smoothsketch: 3D free-form shapes from complex sketches. ACM Trans. Graph. 25(3), 589–598 (2006)CrossRefGoogle Scholar
  9. 9.
    Karpenko, O.A., Hughes, J.F., Raskar, R.: Free-form sketching with variational implicit surfaces. Comput. Graph. Forum 21(3), 585–594 (2002)CrossRefGoogle Scholar
  10. 10.
    Letniowski, F.W.: Three-dimensional delaunay triangulations for finite element approximations to a second-order diffusion operator. SIAM J. Sci. Stat. Comput. 13, 765–772 (1992)CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    Mäntylä, M.: Boolean operations of 2-manifolds through vertex neighborhood classification. ACM Trans. Graph. 5(1), 1–29 (1986)CrossRefGoogle Scholar
  12. 12.
    Nealen, A., Igarashi, T., Sorkine, O., Alexa, M.: FiberMesh: designing freeform surfaces with 3D curves. In: ACM SIGGRAPH, p. 41. ACM, New York, NY (2007)Google Scholar
  13. 13.
    Owada, S., Nielsen, F., Nakazawa, K., Igarashi, T.: A sketching interface for modeling the internal structures of 3D shapes. In: Proceedings of Smart Graphics 2003. Springer, Heidelberg (2003)Google Scholar
  14. 14.
    Shewchuk, J.R.: Triangle: Engineering a 2d quality mesh generator and delaunay triangulator. In: First Workshop on Applied Computational Geometry, pp. 124–133. ACM Press, Philadelphia (1996)Google Scholar
  15. 15.
    Sukumar, N.: Voronoi cell finite difference method for the diffusion operator on arbitrary unstructured grids. Int. J. Numer. Methods Eng. 57, 1–34 (2003)CrossRefMathSciNetzbMATHGoogle Scholar
  16. 16.
    Tai, C.L., Zhang, H., Fong, J.C.K.: Prototype modeling from sketched silhouettes based on convolution surfaces. Comput. Graph. Forum 23(1), 71–83 (2005)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of Computer Science and EngineeringThe Chinese University of Hong KongHong KongP.R. China
  2. 2.University of Macau & Inst. of SoftwareChinese Academy of SciencesMacauP.R. China
  3. 3.The Chinese University of Hong KongShatinP.R. China

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