article

Discrete conformal mappings via circle patterns

Authors:

Liliya Kharevych,

Boris Springborn,

Peter SchröderAuthors Info & Claims

ACM Transactions on Graphics (TOG), Volume 25, Issue 2

Pages 412 - 438

https://doi.org/10.1145/1138450.1138461

Published: 01 April 2006 Publication History

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.]]

Digital Library

[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.]]

Digital Library

[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.]]

Digital Library

[10]

Erickson, J. and Har-Peled, S. 2004. Optimally cutting a surface into a disk. Discrete Computat. Geometry 31, 1, 37--59.]]

Digital Library

[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.]]

Digital Library

[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.]]

Digital Library

[15]

Gu, X. and Yau, S.-T. 2003. Global conformal surface parameterization. In Proceedings of the Symposium on Geometry Processing. Eurographics Association, 127--137.]]

Digital Library

[16]

Hirani, A. N. 2003. Discrete exterior calculus. Ph.D. thesis. California Institute of Technology.]]

Digital Library

[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.]]

Digital Library

[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.]]

Digital Library

[19]

Katz, S. and Tal, A. 2003. Hierarchical mesh decomposition using fuzzy clustering and cuts. ACM Trans. Graph. 22, 3, 954--961.]]

Digital Library

[20]

Khodakovsky, A., Litke, N., and Schröder, P. 2003. Globally smooth parameterizations with low distortion. ACM Trans. Graph. 22, 3, 350--357.]]

Digital Library

[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.]]

Digital Library

[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.]]

Digital Library

[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.]]

Digital Library

[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.]]

Digital Library

[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.]]

Digital Library

[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.]]

Digital Library

[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.]]

Digital Library

[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.]]

Digital Library

[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.]]

Cited By

Liu SJi YGuo JLiu LFu X(2024)Smooth Bijective Projection in a High-order ShellACM Transactions on Graphics10.1145/365820743:4(1-13)Online publication date: 19-Jul-2024
https://dl.acm.org/doi/10.1145/3658207
Capouellez RZorin D(2024)Seamless Parametrization in Penner CoordinatesACM Transactions on Graphics10.1145/365820243:4(1-13)Online publication date: 19-Jul-2024
https://dl.acm.org/doi/10.1145/3658202
Zawallich LPajarola R(2024)Unfolding via Mesh Approximation using Surface FlowsComputer Graphics Forum10.1111/cgf.1503143:2Online publication date: 27-Apr-2024
https://doi.org/10.1111/cgf.15031
Show More Cited By

Index Terms

Discrete conformal mappings via circle patterns
1. Computing methodologies
  1. Computer graphics
    1. Shape modeling
2. Theory of computation
  1. Randomness, geometry and discrete structures
    1. Computational geometry

Recommendations

Conformal equivalence of triangle meshes

We present a new algorithm for conformal mesh parameterization. It is based on a precise notion of discrete conformal equivalence for triangle meshes which mimics the notion of conformal equivalence for smooth surfaces. The problem of finding a flat ...
Discrete conformal mappings via circle patterns
SIGGRAPH '05: ACM SIGGRAPH 2005 Courses

We introduce a novel method for the construction of discrete conformal mappings from (regions of) embedded meshes to the plane. Our approach is based on circle patterns, i.e., arrangements of circles---one for each face---with prescribed intersection ...
Robust fairing via conformal curvature flow

We present a formulation of Willmore flow for triangulated surfaces that permits extraordinarily large time steps and naturally preserves the quality of the input mesh. The main insight is that Willmore flow becomes remarkably stable when expressed in ...

Comments

Information & Contributors

Information

Published In

cover image ACM Transactions on Graphics

ACM Transactions on Graphics Volume 25, Issue 2

April 2006

288 pages

ISSN:0730-0301

EISSN:1557-7368

DOI:10.1145/1138450

Issue’s Table of Contents

Copyright © 2006 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 01 April 2006

Published in TOG Volume 25, Issue 2

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Article

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

237
Total Citations
View Citations
1,598
Total Downloads

Downloads (Last 12 months)72
Downloads (Last 6 weeks)6

Reflects downloads up to 06 Oct 2024

Other Metrics

View Author Metrics

Citations

Cited By

Liu SJi YGuo JLiu LFu X(2024)Smooth Bijective Projection in a High-order ShellACM Transactions on Graphics10.1145/365820743:4(1-13)Online publication date: 19-Jul-2024
https://dl.acm.org/doi/10.1145/3658207
Capouellez RZorin D(2024)Seamless Parametrization in Penner CoordinatesACM Transactions on Graphics10.1145/365820243:4(1-13)Online publication date: 19-Jul-2024
https://dl.acm.org/doi/10.1145/3658202
Zawallich LPajarola R(2024)Unfolding via Mesh Approximation using Surface FlowsComputer Graphics Forum10.1111/cgf.1503143:2Online publication date: 27-Apr-2024
https://doi.org/10.1111/cgf.15031
Genest BCourty NCoeurjolly D(2024)Non‐Euclidean Sliced Optimal Transport SamplingComputer Graphics Forum10.1111/cgf.1502043:2Online publication date: 30-Apr-2024
https://doi.org/10.1111/cgf.15020
Zhang ZLi MZhao ZFang QFu X(2024)Practical Integer-Constrained Cone Construction for Conformal ParameterizationsIEEE Transactions on Visualization and Computer Graphics10.1109/TVCG.2023.328730330:8(5227-5239)Online publication date: Aug-2024
https://doi.org/10.1109/TVCG.2023.3287303
Choi G(2024)Fast ellipsoidal conformal and quasi-conformal parameterization of genus-0 closed surfacesJournal of Computational and Applied Mathematics10.1016/j.cam.2024.115888447(115888)Online publication date: Sep-2024
https://doi.org/10.1016/j.cam.2024.115888
Dai YSu JFu X(2024)Anisotropic triangular meshing using metric-adapted embeddingsComputer Aided Geometric Design10.1016/j.cagd.2024.102314111:COnline publication date: 1-Jun-2024
https://dl.acm.org/doi/10.1016/j.cagd.2024.102314
Li MFang QZhang ZLiu LFu X(2023)Efficient Cone Singularity Construction for Conformal ParameterizationsACM Transactions on Graphics10.1145/361840742:6(1-13)Online publication date: 5-Dec-2023
https://dl.acm.org/doi/10.1145/3618407
Capouellez RZorin D(2023)Metric Optimization in Penner CoordinatesACM Transactions on Graphics10.1145/361839442:6(1-19)Online publication date: 5-Dec-2023
https://dl.acm.org/doi/10.1145/3618394
Bommes DZimmer HKobbelt L(2023)Mixed-Integer QuadrangulationSeminal Graphics Papers: Pushing the Boundaries, Volume 210.1145/3596711.3596740(249-258)Online publication date: 1-Aug-2023
https://dl.acm.org/doi/10.1145/3596711.3596740
Show More Cited By

View Options

Get Access

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Article

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Media

Figures

Other

Tables

View Issue’s Table of Contents