Constraints on Curve Networks Suitable for G2 Interpolation

  • Thomas Hermann
  • Jorg Peters
  • Tim Strotman
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6130)


When interpolating a network of curves to create a C 1 surface from smooth patches, the network has to satisfy an algebraic condition, called the vertex enclosure constraint. We show the existence of an additional constraint that governs the admissibility of curve networks for G 2 interpolation by smooth patches.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [DS91]
    Du, W.-H., Schmitt, F.J.M.: G 1 smooth connection between rectangular and triangular Bézier patches at a common corner. In: Laurent, P.-J., Le Méhauté, A., Schumaker, L.L. (eds.) Curves and Surfaces, pp. 161–168. Academic Press, London (1991)Google Scholar
  2. [DS92]
    Du, W.-H., Schmitt, F.J.M.: On the G 2 continuity of piecewise parametric surfaces. In: Lyche, S. (ed.) Mathematical Methods in CAGD II, pp. 197–207 (1992)Google Scholar
  3. [GH95]
    Grimm, C.M., Hughes, J.F.: Modeling surfaces of arbitrary topology using manifolds. In: Computer Graphics. Annual Conference Series, vol. 29, pp. 359–368 (1995)Google Scholar
  4. [Gre74]
    Gregory, J.A.: Smooth interpolation without twist constraints. In: Barnhill, R.E., Riesenfeld, R.F. (eds.) Computer Aided Geometric Design, pp. 71–87. Academic Press, London (1974)Google Scholar
  5. [Hah89]
    Hahn, J.: Filling polygonal holes with rectangular patches. In: Theory and practice of geometric modeling (Blaubeuren, 1988), pp. 81–91. Springer, Berlin (1989)Google Scholar
  6. [Her96]
    Hermann, T.: G 2 interpolation of free form curve networks by biquintic Gregory patches. Computer Aided Geometric Design 13, 873–893 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  7. [HLW99]
    Hermann, T., Lukács, G., Wolter, F.E.: Geometrical criteria on the higher order smoothness of composite surfaces. Computer Aided Geometric Design 17, 907–911 (1999)CrossRefGoogle Scholar
  8. [HPS09]
    Hermann, T., Peters, J., Strotman, T.: A geometric criterion for smooth interpolation of curve networks. In: Keyser, J. (ed.) SPM 2009: 2009 SIAM/ACM Joint Conference on Geometric and Physical Modeling, pp. 169–173. ACM, New York (2009)CrossRefGoogle Scholar
  9. [KP09]
    Karčiauskas, K., Peters, J.: Guided spline surfaces. Computer Aided Geometric Design 26(1), 105–116 (2009)CrossRefMathSciNetGoogle Scholar
  10. [LS08]
    Loop, C.T., Schaefer, S.: G 2 tensor product splines over extraordinary vertices. Comput. Graph. Forum 27(5), 1373–1382 (2008)CrossRefGoogle Scholar
  11. [MW91]
    Miura, K.T., Wang, K.K.: C 2 Gregory patch. In: Post, F.H., Barth, W. (eds.) EUROGRAPHICS 1991, pp. 481–492. North-Holland, Amsterdam (1991)Google Scholar
  12. [PBP02]
    Prautzsch, H., Boehm, W., Paluzny, M.: Bézier and B-Spline Techniques. Springer, Heidelberg (2002)zbMATHGoogle Scholar
  13. [Pet91]
    Peters, J.: Smooth interpolation of a mesh of curves. Constructive Approximation 7, 221–246 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  14. [Pet92]
    Peters, J.: Joining smooth patches at a vertex to form a C k surface. Computer-Aided Geometric Design 9, 387–411 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  15. [Pet02]
    Peters, J.: Geometric continuity. In: Handbook of Computer Aided Geometric Design, pp. 193–229. Elsevier, Amsterdam (2002)CrossRefGoogle Scholar
  16. [Pra97]
    Prautzsch, H.: Freeform splines. Computer Aided Geometric Design 14(3), 201–206 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  17. [Rei98]
    Reif, U.: TURBS—topologically unrestricted rational B-splines. Constructive Approximation 14(1), 57–77 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  18. [Ye97]
    Ye, X.: Curvature continuous interpolation of curve meshes. Computer Aided Geometric Design 14(2), 169–190 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  19. [YZ04]
    Ying, L., Zorin, D.: A simple manifold-based construction of surfaces of arbitrary smoothness. ACM TOG 23(3), 271–275 (2004)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Thomas Hermann
    • 1
  • Jorg Peters
    • 2
  • Tim Strotman
    • 1
  1. 1.Parasolid Components, Siemens PLM Software 
  2. 2.University of Florida 

Personalised recommendations