Variations on Angle Based Flattening

  • Rhaleb Zayer
  • Christian Rössl
  • Hans-Peter Seidel
Conference paper
Part of the Mathematics and Visualization book series (MATHVISUAL)


Angle Based Flattening is a robust parameterization technique allowing a free boundary. The numerical optimisation associated with the approach yields a challenging problem. We discuss several approaches to effectively reduce the computational effort involved and propose appropriate numerical solvers. We propose a simple but effective transformation of the problem which reduces the computational cost and simplifies the implementation. We also show that fast convergence can be achieved by finding approximate solutions which yield a low angular distortion.


Inequality Constraint Boundary Control Texture Mapping Iterative Solver Angular Distortion 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    D.P. Bertsekas. Constrained optimization and lagrange multiplier methods. Athena Scientific, 1996.Google Scholar
  2. 2.
    R.H. Byrd and J. Nocedal. Active set and interior methods for nonlinear optimization Doc. MATH, Extra Volume ICM III, 1998, pp. 667–676.MathSciNetGoogle Scholar
  3. 3.
    M. Desbrun, M. Meyer, and P. Alliez. Intrinsic parameterizations of triangle meshes. Proc. Eurographics 2002, pp. 209–218.Google Scholar
  4. 4.
    G. Di Battista and L. Vismara. Angles of planar triangular graphs. SIAM Journal on Discrete Mathematics, 9(3), 1996, pp. 349–359.MathSciNetCrossRefGoogle Scholar
  5. 5.
    M. Eck, T. DeRose, T. Duchamp, H. Hoppe, M. Lounsbery, and W. Stuetzle. Multiresolution analysis of arbitrary meshes. Proc. ACM SIGGRAPH '95, pp. 173–182.Google Scholar
  6. 6.
    M. S. Floater. Parametrization and smooth approximation of surface triangulations. Comp. Aided Geom. Design, (14), 3, 1997, pp. 231–250.zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    M. S. Floater and K. Hormann. Parameterization of triangulations and unorganized points. Tutorials on Multiresolution in Geometric Modelling Springer-Verlag, Heidelberg (2002), pp. 287–315.Google Scholar
  8. 8.
    M. S. Floater. Mean value coordinates. Comp. Aided Geom. Design, (20), 1, 2003, pp. 19–27.zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    M. S. Floater and K. Hormann. Surface parameterization: a tutorial and survey, Advances in Multiresolution for Geometric Modelling, N. A. Dodgson, M. S. Floater, and M. A. Sabin (eds.), Springer, 2004, pp. 157–186 (this book).Google Scholar
  10. 10.
    M. R. Hestenes and E. Stiefel. Method of conjugate gradients for solving linear systems. J. Res. Nat. Bur. Stand. 49:409–436, 1952.MathSciNetGoogle Scholar
  11. 11.
    A. Garg. New results on drawing angle graphs. Computational Geometry (9), (1–2), 1998, pp. 43–82.zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    A. Greenbaum. Iterative Methods for Solving Linear Systems SIAM, Philadelphia, 1997.Google Scholar
  13. 13.
    S. Haker, S. Angenent, A. Tannenbaum, R. Kikinis, G. Sapiro, and M. Halle. Conformal surface parameterization for texture mapping. IEEE Transactions on Visualization and Computer Graphics, 6(2), 2000, pp. 181–189.CrossRefGoogle Scholar
  14. 14.
    K. Hormann and G. Greiner. MIPS: an efficient global parametrization method. Curve and Surface Design: Saint-Malo 1999, 2000, pp. 153–162.Google Scholar
  15. 15.
    B. Levy, S. Petitjean, N. Ray, and J. Maillot. Least squares conformal maps for automatic texture atlas generation. Proc. ACM SIGGRAPH 2002, pp. 362–371.Google Scholar
  16. 16.
    J. Liesen, E. de Sturler, A. Sheffer, Y. Aydin, and C. Siefert. Preconditioners for indefinite linear systems arising in surface parameterization. Proceedings of the 10th International Meshing Round Table, 2001, pp. 71–81.Google Scholar
  17. 17.
    C. Paige and M. Saunders. Solution of sparse indefinite systems of linear equations. SIAM J. Numer. Anal, 12, 1975, pp. 617–629.MathSciNetCrossRefGoogle Scholar
  18. 18.
    U. Pinkall and K. Polthier. Computing discrete minimal surfaces and their conjugates. Experimental Mathematics, 2(15), 1993, pp. 15–36.MathSciNetGoogle Scholar
  19. 19.
    P. Sander, J. Snyder, S. Gortler, and H. Hoppe. Texture mapping progressive meshes. Proc. ACM SIGGRAPH 2001, pp. 409–416.Google Scholar
  20. 20.
    A. Sheffer and E. de Sturler. Parameterization of faceted surfaces for meshing using angle based flattening. Engineering with Computers, 17(3), 2001, pp. 326–337.CrossRefGoogle Scholar
  21. 21.
    A. Sheffer and E. de Sturler. Smoothing an overlay grid to minimize linear distortion in texture mapping. ACM Transactions on Graphics, 21(4), 2002.Google Scholar
  22. 22.
    O. Sorkine, D. Cohen-Or, R. Goldenthal, and D. Lischinski. Bounded-distortion piecewise mesh parameterization. Proc. IEEE Visualization 2002, pp. 355–362.Google Scholar
  23. 23.
    W. T. Tutte. How to draw a graph. Proceedings of the London Mathematical Society, 13(3), 1963, pp. 743–768.zbMATHMathSciNetGoogle Scholar
  24. 24.
    G. Zigelmann, R. Kimmel, and N. Kiryati. Texture mapping using surface flattening via multi-dimensional scaling. IEEE Transactions on Visualization and Computer Graphics, 8(2), 2002.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Rhaleb Zayer
    • 1
  • Christian Rössl
    • 1
  • Hans-Peter Seidel
    • 1
  1. 1.Max-Planck-Institut für InformatikSaarbrückenGermany

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