Matheon      

DFG Research Center Matheon “Mathematics for key technologies”
Project F1: Discrete Surface Parametrizations

       TU Berlin
   

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Events

Matheon

TU Geometry

TU Discrete Geometry



     

People

Head Alexander I. Bobenko Günter M. Ziegler
Members Ulrike Bücking (née Scheerer) Stefan Sechelmann Boris Springborn
Former members Yuri Suris Sergey Tsarev Thorsten Theobald

Research

The aim of discrete differential geometry is the discretization of classical differential geometry, that is, to find proper discrete analogs of differential geometric notions and to develop at the discrete level a corresponding theory. The classical theory of surfaces relies on special parametrizations, such as curvature line parametrizations or conformal parametrizations. In applications, surfaces are often described by surface meshes. In this project we search for practically relevant and theoretically satisfying answers to questions like, for example: “What does it mean for a surface mesh to be a curvature line parametrization?”

Publications

A. I. Bobenko, Yu. B. Suris. Discrete Differential Geometry: Integrable Structure. Graduate Studies in Mathematics, vol. 98, Amer. Math. Soc., Providence, RI, 2008, 404 pages.
B. Springborn, P. Schröder, U. Pinkall. Conformal equivalence of triangle meshes. ACM Transactions on Graphics 27:3 (2008). [Proceedings of ACM SIGGRAPH 2008].
G. M. Ziegler. Nonrational configurations, polytopes, and surfaces. Mathematical Intelligencer 30:3 (2008) 36-42.
U. Bücking. Approximation of conformal mappings by circle patterns . Geom. Dedicata 173:1 (2008) 163-197.
B. Springborn. A variational principle for weighted Delaunay triangulations and hyperideal polyhedra. J. Differential Geom. 78 (2008) 333-367.
A. I. Bobenko, P. Schröder, J. M. Sullivan, G. M. Ziegler, editors. Discrete Differential Geometry. Oberwolfach Seminars, vol. 38, Birkhäuser, Basel, 2008, 341 pages.
G. M. Ziegler. Polyhedral surfaces of high genus. Preprint, 2005. In: A. I. Bobenko, John M. Sullivan, Peter Schröder (editors). Discrete Differential Geometry. Oberwolfach Seminars vol. 38, Birkhäuser, Basel, 2008, pp. 191-213.
U. Pinkall, B. Springborn, S. Weißmann. A new doubly discrete analogue of smoke ring flow and the real time simulation of fluid flow. J. Phys. A: Math. Theor. 40 (2007), 12563-12576.
M. Fisher, B. Springborn, P. Schröder, A. I. Bobenko. An algorithm for the construction of intrinsic delaunay triangulations with applications to digital geometry processing. Computing 81 (2007) 199-213.
A. I. Bobenko, B. Springborn. A discrete Laplace-Beltrami operator for simplicial surfaces. Discrete Comput. Geom. 38:4 (2007) 740-756.
G. M. Ziegler (with an appendix by Th. Schröder and N. Witte). Convex Polytopes: Extremal constructions and f-vector shapes. In: E. Miller, V. Reiner, and B. Sturmfels, editors, Geometric Combinatorics, Proc. Park City Mathematical Institute (PCMI) 2004, Amer. Math. Soc., Providence, RI, 2007, pages 617-691.
A. I. Bobenko, Tim Hoffmann, B. Springborn. Minimal surfaces from circle patterns. Geometry from combinatorics. Ann. of Math. 164:1 (2006), 231-264.
Liliya Kharevych, B. Springborn, Peter Schröder. Discrete conformal mappings via circle patterns. ACM Transactions on Graphics 25:2 (2006), 412-438 .
A. I. Bobenko. Geometry of discrete integrability. The consistency approach, pp. 43-53 in Faddeev et al. (eds.), Bilinear Integrable Systems: From Classical to Quantum, Continuous to Discrete. Springer, 2006.
B. Springborn. A unique representation of polyhedral types. Centering via Möbius transformations. Math. Z. 249:3 (2005), 513 - 517.
A. I. Bobenko, Peter Schröder. Discrete Willmore flow. Eurographics Symposium on Geometry Processing 2005, pp. 101-110.
A.I. Bobenko, Christian Mercat, Yu. Suris. Linear and nonlinear theories of discrete analytic functions. Integrable structure and isomonodromic Green's function. J. Reine Angew. Math. 583 (2005). 117-161.
A. I. Bobenko. A conformal energy for simplicial surfaces, pp. 133-143 in J. E. Goodman, J. Pach, Emo Welzl (eds.), Combinatorial and Computational Geometry, MSRI Publications Vol. 52, Cambridge University Press, 2005.
A. I. Bobenko, Daniel Matthes, Yu. B. Suris. Nonlinear hyperbolic equations in surface theory: integrable discretizations and approximation results. Algebra and Analysis 17:1 (2005), 53-83.
A. I. Bobenko, B. Springborn. Variational principles for circle patterns and Koebe's theorem. Trans. Amer. Math. Soc. 356 (2004), 659-689.
A. I. Bobenko. Discrete Differential Geometry. Integrability as Consistency, pp. 85-110 in B. Grammaticos, T. Tamizhmani, Y. Kosmann-Schwarzbach (eds.), Discrete Integrable Systems, Lecture Notes in Physics Vol. 644, Springer, 2004.
Vsevolod E. Adler, A. I. Bobenko, Yu. B. Suris. Geometry of Yang-Baxter maps: pencils of conics and quadrirational mappings. Communications in Analysis and Geometry 12:5 (2004), 967-1008.
A. I. Bobenko, Daniel Matthes, Yu. B. Suris. Discrete and smooth orthogonal systems: C∞-approximation. Internat. Math. Research Notices 2003:45 (2003), 2415-2459.
Sergey I. Agafonov, A. I. Bobenko. Hexagonal circle patterns with constant intersection angles and discrete Painleve and Riccati equations. J. Math. Phys. 44:8 (2003), 3455-3469.
Boris Springborn . 27.02.2009. Accesses: 27159 since 11.03
DFG Research Center Matheon "Mathematics for key technologies"