Discrete conformal mappings via circle patterns

Abstract

We introduce a novel method for the construction of discrete conformal mappings from surface meshes of arbitrary topology to the plane. Our approach is based on circle patterns, that is, arrangements of circles---one for each face---with prescribed intersection angles. Given these angles, the circle radii follow as the unique minimizer of a convex energy. The method supports very flexible boundary conditions ranging from free boundaries to control of the boundary shape via prescribed curvatures. Closed meshes of genus zero can be parameterized over the sphere. To parameterize higher genus meshes, we introduce cone singularities at designated vertices. The parameter domain is then a piecewise Euclidean surface. Cone singularities can also help to reduce the often very large area distortion of global conformal maps to moderate levels. Our method involves two optimization problems: a quadratic program and the unconstrained minimization of the circle pattern energy. The latter is a convex function of logarithmic radius variables with simple explicit expressions for gradient and Hessian. We demonstrate the versatility and performance of our algorithm with a variety of examples.

References

1. Bern, M. and Eppstein, D. 2001. Optimal Möbius transformations for information visualization and meshing. In Algorithms and Data Structures. Providence, RI. Lecture Notes in Computer Science, vol. 2125. Springer-Verlag, Berlin, Germany, 14--25.]]
2. Bobenko, A. I. and Springborn, B. A. 2004. Variational principles for circle patterns and Koebe's Theorem. Trans. Amer. Math. Soc. 356, 659--689.]]
3. Bobenko, A. I. and Springborn, B. A. 2005. A discrete Laplace-Beltrami operator for simplicial surfaces. http://arxiv.org/abs/math.DG/0503219.]]
4. Bowers, P. L. and Hurdal, M. K. 2003. Planar conformal mappings of piecewise flat surfaces. In Visualization and Mathematics III, H.-C. Hege and K. Polthier, Eds. Mathematics and Visualization. Springer-Verlag, Berlin, Germany, 3--34.]]
5. Brägger, W. 1992. Kreispackungen und Triangulierungen. L'Enseignement Mathématique 38, 201--217.]]
6. Colin de Verdière, Y. 1991. Un principe variationnel pour les empilements de cercles. Inventiones Mathematicae 104, 655--669.]]
7. Collins, C. and Stephenson, K. 2003. A circle packing algorithm. Computa. Geometry: Theory Appl. 25, 233--256.]]
8. Desbrun, M., Meyer, M., and Alliez, P. 2002. Intrinsic parameterizations of surface meshes. In Proceedings of Eurographics Computer Graphics Forum 21, 3, 209--218.]]
9. Dillencourt, M. B. and Smith, W. D. 1996. Graph-theoretical conditions for inscribability and Delaunay realizability. Discrete Math. 161, 63--77.]]
10. Erickson, J. and Har-Peled, S. 2004. Optimally cutting a surface into a disk. Discrete Computat. Geometry 31, 1, 37--59.]]
11. Friedel, I., Schröder, P., and Desbrun, M. 2005. Unconstrained spherical parameterization. (http://multires.caltech.edu/pubs/SParam_JGT.pdf). Submitted for Publication.]]
12. Garland, M., Willmott, A., and Heckbert, P. S. 2001. Hierarchical face clustering on polygonal surfaces. In Proceedings of the Symposium on Interactive 3D Graphics. ACM Press, 49--58.]]
13. Grünbaum, B. 2003. Convex Polytopes, 2nd Ed. Graduate Texts in Mathematics, vol. 221. Springer-Verlag, New York, NY.]]
14. Gu, X., Gortler, S. J., and Hoppe, H. 2002. Geometry images. ACM Trans. Graph. 21, 3, 355--361.]]
15. Gu, X. and Yau, S.-T. 2003. Global conformal surface parameterization. In Proceedings of the Symposium on Geometry Processing. Eurographics Association, 127--137.]]
16. Hirani, A. N. 2003. Discrete exterior calculus. Ph.D. thesis. California Institute of Technology.]]
17. Indermitte, C., Liebling, T., Troyanov, M., and Clémençon, H. 2001. Voronoi diagrams on piecewise flat surfaces and an application to biological growth. Theoret. Comput. Science 263, 1--2, 268--274.]]
18. Jin, M., Wang, Y., Yau, S.-T., and Gu, X. 2004. Optimal global conformal surface parameterizations. In Proceedings of IEEE Visualization. IEEE Computer Society, 267--274.]]
19. Katz, S. and Tal, A. 2003. Hierarchical mesh decomposition using fuzzy clustering and cuts. ACM Trans. Graph. 22, 3, 954--961.]]
20. Khodakovsky, A., Litke, N., and Schröder, P. 2003. Globally smooth parameterizations with low distortion. ACM Trans. Graph. 22, 3, 350--357.]]
21. Leibon, G. 2002. Characterizing the Delaunay decompositions of compact hyperbolic surfaces. Geometry Topolo. 6, 361--391.]]
22. Lévy, B., Petitjean, S., Ray, N., and Maillot, J. 2002. Least squares conformal maps for automatic texture atlas generation. ACM Trans. Graph. 21, 3, 362--371.]]
23. Mercat, C. 2001. Discrete Riemann surfaces and the Ising model. Comm. Math. Physics 218, 1, 177--216.]]
24. Mosek. 2005. Constrained quadratic minimization software. Version 3.1r42. http://www.mosek.com/.]]
25. Pinkall, U. and Polthier, K. 1993. Computing discrete minimal surfaces and their conjugates. Experim. Math. 2, 1, 15--36.]]
26. Ray, N., Li, W. C., Lévy, B., Sheffer, A., and Alliez, P. 2005. Periodic global parameterizations. (http://www.loria.fr/~levy/publications/papers/2005/PGP/pgp.pdf). Submitted for publication.]]
27. Rivin, I. 1994. Euclidean structures on simplicial surfaces and hyperbolic volume. Annals of Math. 139, 3, 553--580.]]
28. Rodin, B. and Sullivan, D. 1987. The convergence of circle packings to the Riemann mapping. J. Differ. Geometry 26, 2, 349--360.]]
29. Sander, P. V., Snyder, J., Gortler, S. J., and Hoppe, H. 2001. Texture mapping progressive meshes. In Proceedings of SIGGRAPH. 409--416.]]
30. Sander, P. V., Wood, Z. J., Gortler, S. J., Snyder, J., and Hoppe, H. 2003. Multi-chart geometry images. In Proceedings of the Symposium on Geometry Processing. Eurographics Association, 146--155.]]
31. Sheffer, A. and de Sturler, E. 2000. Surface Parameterization for meshing by triangulation flattening. In Proceedings of the 9th International Meshing Roundtable. Sandia National Labs, 161--172.]]
32. Sheffer, A. and Hart, J. C. 2002. Seamster: Inconspicuous low-distortion texture seam layout. In Proceedings of IEEE Visualization. IEEE Computer Society, 291--298.]]
33. Sheffer, A., Lévy, B., Mogilnitski, M., and Bogomyakov, A. 2004. ABF++: fast and robust angle based flattening. ACM Trans. Graph. 24, 2, 311--330.]]
34. Sorkine, O., Cohen-Or, D., Goldenthal, R., and Lischinski, D. 2002. Bounded-distortion piecewise mesh parameterization. In Proceedings of IEEE Visualization. IEEE Computer Society, 355--362.]]
35. Springborn, B. A. 2005. A unique representation of polyhedral types. Centering via Möbius transformations. Mathematische Zeitschrift 249, 3, 513--517.]]
36. Tarini, M., Hormann, K., Cignoni, P., and Montani, C. 2004. PolyCube-Maps. ACM Trans. Graph. 23, 3, 853--860.]]
37. Thurston, W. P. 1980. The geometry and topology of three-manifolds. Available at http://www.msri.org/publications/books/gt3m/.]]
38. Thurston, W. P. 1985. The finite Riemann mapping theorem. At the Symposium on the Occasion of the Proof of the Bieberbach Conjecture. Purdue University (March).]]
39. Troyanov, M. 1991. Prescribing curvature on compact surfaces with conical singularities. Trans. Amer. Math. Society 324, 793--821.]]
40. Zayer, R., Rössl, C., and Seidel, H.-P. 2003. Convex boundary angle based flattening. In Proceedings of Vision, Modeling and Visualiza. 281--288.]]
41. Zhang, E., Mischaikow, K., and Turk, G. 2005. Feature-based surface parameterization and texture mapping. ACM Trans. Graph. 24, 1, 1--27.]]
42. Ziegler, G. M. 2004. Convex polytopes: Extremal constructions and f-vector shapes. IAS/Park City Mathematics Series, vol. 14. http://arxiv.org/abs/math/0411400. To Appear.]]

Index Terms

• Computing methodologies

  • Computer graphics

    • Shape modeling

• Theory of computation

  • Randomness, geometry and discrete structures

    • Computational geometry