Abstract
The notion of discrete conformality proposed by Luo (2004) and Bobenko et al. (2015) on triangle meshes has rich mathematical theories and wide applications. Gu et al. (2019) and Wu and Zhu (2020) proved that the discrete uniformizations approximate the continuous uniformization for closed surfaces of genus \(\ge 1\), given that the approximating triangle meshes are reasonably good. In this paper, we generalize this result to the remaining case of genus zero surfaces, by reducing it to planar cases via stereographic projections.
Similar content being viewed by others
References
Bobenko, A.I., Pinkall, U., Springborn, B.A.: Discrete conformal maps and ideal hyperbolic polyhedra. Geom. Topol. 19(4), 2155–2215 (2015)
Bücking, U.: Approximation of conformal mappings using conformally equivalent triangular lattices. In: Advances in Discrete Differential Geometry, pp. 133–149. Springer, Berlin (2016)
Bücking, U.: \(C^\infty \)-convergence of conformal mappings for conformally equivalent triangular lattices. Results Math. 73(2), # 84 (2018)
Devadoss, S.L., O’Rourke, J.: Discrete and Computational Geometry. Princeton University Press, Princeton (2011)
Gu, D., Luo, F., Wu, T.: Convergence of discrete conformal geometry and computation of uniformization maps. Asian J. Math. 23(1), 21–34 (2019)
Luo, F.: Combinatorial Yamabe flow on surfaces. Commun. Contemp. Math. 6(5), 765–780 (2004)
Luo, F., Sun, J., Wu, T.: Discrete conformal geometry of polyhedral surfaces and its convergence. Geom. Topol. 26(3), 937–987 (2022)
Pinkall, U., Polthier, K.: Computing discrete minimal surfaces and their conjugates. Exp. Math. 2(1), 15–36 (1993)
Rivin, I.: Intrinsic geometry of convex ideal polyhedra in hyperbolic \(3\)-space. In: Analysis, Algebra, and Computers in Mathematical Research (Luleå 1992). Lecture Notes in Pure and Appl. Math., vol. 156, pp. 275–291. Dekker, New York (1994)
Rodin, B., Sullivan, D.: The convergence of circle packings to the Riemann mapping. J. Differ. Geom. 26(2), 349–360 (1987)
Springborn, B.: Ideal hyperbolic polyhedra and discrete uniformization. Discrete Comput. Geom. 64(1), 63–108 (2020)
Springborn, B., Schröder, P., Pinkall, U.: Conformal equivalence of triangle meshes. ACM Trans. Graph. 27(3), 1–11 (2008)
Wu, T., Zhu, X.: The convergence of discrete uniformizations for closed surfaces (2020). arXiv:2008.06744
Acknowledgements
The work is supported in part by NSF 1719582, NSF 1760471, NSF 1760527, NSF DMS 1737876, and NSF 1811878.
Author information
Authors and Affiliations
Corresponding author
Additional information
Editor in Charge: Kenneth Clarkson
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Luo, Y., Wu, T. & Zhu, X. The Convergence of Discrete Uniformizations for Genus Zero Surfaces. Discrete Comput Geom 71, 1057–1080 (2024). https://doi.org/10.1007/s00454-022-00458-w
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00454-022-00458-w