Morphing Planar Graphs in Spherical Space

  • Stephen G. Kobourov
  • Matthew Landis
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4372)


We consider the problem of intersection-free planar graph morphing, and in particular, a generalization from Euclidean space to spherical space. We show that there exists a continuous and intersection-free morph between two sphere drawings of a maximally planar graph, provided that both sphere drawings have convex inscribed polytopes, where sphere drawings are the spherical equivalent of plane drawings: intersection-free geodesic-arc drawings. In addition, we describe a morphing algorithm along with its implementation. Movies of sample morphs can be found at


  1. 1.
    Alfeld, P., Neamtu, M., Schumaker, L.L.: Bernstein-Bézier polynomials on circle, spheres, and sphere-like surfaces. Computer Aided Geometric Design Journal 13, 333–349 (1996)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Aronov, B., Seidel, R., Souvaine, D.: On compatible triangulations of simple polygons. Computational Geometry: Theory and Applications 3, 27–35 (1993)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Biedl, T.C., Lubiw, A., Spriggs, M.J.: Morphing planar graphs while preserving edge directions. In: Healy, P., Nikolov, N.S. (eds.) GD 2005. LNCS, vol. 3843, pp. 13–24. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  4. 4.
    Cairns, S.S.: Deformations of plane rectilinear complexes. American Math. Monthly 51, 247–252 (1944)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Coxeter, H.: Introduction to Geometry. Wiley, Chichester (1961)MATHGoogle Scholar
  6. 6.
    Erten, C., Kobourov, S.G., Pitta, C.: Intersection-free morphing of planar graphs. In: Liotta, G. (ed.) GD 2003. LNCS, vol. 2912, pp. 320–331. Springer, Heidelberg (2004)Google Scholar
  7. 7.
    Floater, M., Gotsman, C.: How to morph tilings injectively. Journal of Computational and Applied Mathematics 101, 117–129 (1999)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Gomes, J., Darsa, L., Costa, B., Vello, D.M.: Warping and Morphing of Graphical Objects. Morgan Kaufmann, San Francisco (1999)Google Scholar
  9. 9.
    Gotsman, C., Gu, X., Sheffer, A.: Fundamentals of spherical parameterization for 3D meshes. In: SIGGRAPH ’03, pp. 358–363 (2003)Google Scholar
  10. 10.
    Gotsman, C., Surazhsky, V.: Guaranteed intersection-free polygon morphing. Computers and Graphics 25(1), 67–75 (2001),, CrossRefGoogle Scholar
  11. 11.
    Hughes, J.F.: Scheduled Fourier volume morphing. Computer Graphics 26(2), 43–46 (1992), CrossRefGoogle Scholar
  12. 12.
    Lubiw, A., Petrick, M., Spriggs, M.: Morphing orthogonal planar graph drawings. In: 17th Symposium on Discrete Algorithms (SODA), pp. 222–230 (2006)Google Scholar
  13. 13.
    Samoilov, T., Elber, G.: Self-intersection elimination in metamorphosis of two-dimensional curves. The Visual Computer 14, 415–428 (1998)CrossRefGoogle Scholar
  14. 14.
    Sederberg, T.W., Greenwood, E.: A physically based approach to 2-D shape blending. In: SIGGRAPH, July 1992, pp. 25–34 (1992)Google Scholar
  15. 15.
    Surazhsky, V., Gotsman, C.: Controllable morphing of compatible planar triangulations. ACM Transactions on Graphics 20(4), 203–231 (2001), doi:10.1145/502783.502784CrossRefGoogle Scholar
  16. 16.
    Thomassen, C.: Deformations of plane graphs. J. Combin. Theory Ser. B 34, 244–257 (1983)MATHMathSciNetGoogle Scholar
  17. 17.
    Tutte, W.T.: How to draw a graph. Proc. London Math. Society 13(52), 743–768 (1963)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Stephen G. Kobourov
    • 1
  • Matthew Landis
    • 1
  1. 1.Department of Computer Science, University of Arizona 

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