Smooth 2D Coordinate Systems on Discrete Surfaces

  • Colin Cartade
  • Rémy Malgouyres
  • Christian Mercat
  • Chafik Samir
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6607)


We introduce a new method to compute conformal parameterizations using a recent definition of discrete conformity, and establish a discrete version of the Riemann mapping theorem. Our algorithm can parameterize triangular, quadrangular and digital meshes. It can be adapted to preserve metric properties. To demonstrate the efficiency of our method, many examples are shown in the experiment section.


parameterization conformal digital surfaces 


  1. 1.
    Ahlfors, L.V.: Complex analysis, 3rd edn. McGraw-Hill Book Co., New York (1978); an introduction to the theory of analytic functions of one complex variable, International Series in Pure and Applied MathematicsGoogle Scholar
  2. 2.
    Bobenko, A.I., Mercat, C., Schmies, M.: Conformal Structures and Period Matrices of Polyhedral Surfaces. In: Computational Approach to Riemann Surfaces. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  3. 3.
    Bobenko, A.I., Schröder, P., Sullivan, J.M., Ziegler, G.M. (eds.): Discrete differential geometry, Oberwolfach Seminars, vol. 38. Birkhäuser, Basel (2008), (papers from the seminar held in Oberwolfach, May 30-June 5, 2004)Google Scholar
  4. 4.
    Desbrun, M., Meyer, M., Alliez, P.: Intrinsic parameterizations of surface meshes. Computer Graphics Forum 21, 209–218 (2002)CrossRefGoogle Scholar
  5. 5.
    Floater, M.S.: Mean value coordinates. Computer Aided Geometric Design 20(1), 19–27 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Fourey, S., Malgouyres, R.: Normals estimation for digital surfaces based on convolutions. Computers & Graphics 33(1), 2–10 (2009)CrossRefzbMATHGoogle Scholar
  7. 7.
    Gu, X.D., Yau, S.T.: Computational conformal geometry, Advanced Lectures in Mathematics (ALM), vol. 3. International Press, Somerville (2008); with 1 CD-ROM (Windows, Macintosh and Linux)Google Scholar
  8. 8.
    Kharevych, L., Springborn, B., Schröder, P.: Discrete conformal mappings via circle patterns. ACM Transactions on Graphics (TOG) 25(2), 438 (2006)CrossRefGoogle Scholar
  9. 9.
    Lévy, B., Petitjean, S., Ray, N., Maillot, J.: Least squares conformal maps for automatic texture atlas generation. ACM Transactions on Graphics 21(3), 362–371 (2002)CrossRefGoogle Scholar
  10. 10.
    Mercat, C.: Discrete Riemann surfaces and the Ising model. Communications in Mathematical Physics 218(1), 177–216 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Mercat, C.: Discrete complex structure on surfel surfaces. In: Coeurjolly, D., Sivignon, I., Tougne, L., Dupont, F. (eds.) DGCI 2008. LNCS, vol. 4992, pp. 153–164. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  12. 12.
    Nocedal, J., Wright, S.J.: Numerical optimization. Springer, Heidelberg (1999)CrossRefzbMATHGoogle Scholar
  13. 13.
    Pinkall, U., Polthier, K.: Computing discrete minimal surfaces and their conjugates. Experimental Mathematics 2(1), 15–36 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Sheffer, A., de Sturler, E.: Parameterization of faceted surfaces for meshing using angle-based flattening. Engineering with Computers 17(3), 326–337 (2001)CrossRefzbMATHGoogle Scholar
  15. 15.
    Tutte, W.: Convex representations of graphs. Proceedings of the London Mathematical Society 3(1), 304 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Wegert, E.: Nonlinear Riemann-Hilbert problems—history and perspectives. In: Computational Methods and Function Theory 1997 (Nicosia), Ser. Approx. Decompos., vol. 11, pp. 583–615. World Sci. Publ., River Edge (1999)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Colin Cartade
    • 1
  • Rémy Malgouyres
    • 1
  • Christian Mercat
    • 2
  • Chafik Samir
    • 3
  1. 1.LIMOSClermont-UniversitéAubièreFrance
  2. 2.IUFMUniversité Lyon 1VileurbanneFrance
  3. 3.ISITClermont-UniversitéAubièreFrance

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