Institute of Geometry

This research area has been explored within the framework of the the National Research Network Industrial Geometry, funded by FWF in 2008-2011. Currently we are exploring Semidiscrete Surfaces (FWF-Project I705, 2012-2016) which is part of the SFB-Transregio Discretization in Geometry and Dynamics, which is funded by DFG and in part by FWF.

Computational Differential Geometry means methods of both numerical and discrete mathematics with the purpose of investigating and modeling curves and surfaces. The main theme of this research project is the robust analysis of differential properties of surfaces, the creation of discrete and semi-discrete models of freeform surfaces, and the study of geometric properties of such models. It is only recently that the wealth of interesting geometry connected to applications in, say, architecture, has come to the attention of mathematicians, and presumably only a small part of it has been investigated.

We are investigating topics of Discrete Differential Geometry: discrete curvatures based on parallel meshes, quad-based and hex-based discrete surfaces, Christoffel duality, and others. New lines of research of semi-discrete surfaces and inverse problems in connection with integral invariants.

The list of publications below also includes work performed before the runtime of the above-mentioned grants.

• J. Wang, C. Jiang, P. Bompas, J. Wallner, and H. Pottmann. Discrete line congruences for shading and lighting. Computer Graphics Forum 32/5 (2013), Proc. Symposium Geometry Processing.
• C. Müller and J. Wallner. Semidiscrete isothermic surfaces. Results Math. 63 (2013), 1395-1407. [doi].
• O. Karpenkov and J. Wallner. On offsets and curvatures for discrete and semidiscrete surfaces. Beitr. Algebra Geom. (2014), to appear. [doi].
• S. Flöry, Y. Nagai, F. Isvoranu, H. Pottmann, and J. Wallner. Ruled free forms. In L. Hesselgren et al., editors, Advances in Architectural Geometry 2012, pages 57-66. Springer, 2013.
• J. Wallner. On the semidiscrete differential geometry of A-surfaces and K-surfaces. J. Geometry 103 (2012), 161-176. [Zbl], [MR], [doi].
• M. Skopenkov, H. Pottmann, and P. Grohs. Ruled Laguerre minimal surfaces. Math. Zeitschrift 272 (2012), 645-674. [doi], [arxiv:1011.0272].
• B. Deng, H. Pottmann, and J. Wallner. Functional webs for freeform architecture. Computer Graphics Forum 30/5 (2011), 1369-1378, Proc. Symp. Geometry Processing. [doi].
• J. Wallner and H. Pottmann. Geometric computing for freeform architecture. J. Math. Industry 1 (2011), #4,1-19. [MR], [doi].
• P. Bo, H. Pottmann, M. Kilian, W. Wang, and J. Wallner. Circular arc structures. ACM Trans. Graphics 30 (2011), #101,1-11, Proc. SIGGRAPH. [doi].
• O. Ebner and S. Haller. Harmonic cohomology of symplectic fiber bundles. Proc. Amer. Math. Soc. 139 (2011), 2827-2931. [MR], [doi], [arxiv:1004.2218].
• H. Pottmann and J. Wallner. Freiformarchitektur und Mathematik. Mitt. DMV 18/2 (2010), 88-95. [Zbl], [MR].
• H. Pottmann, Q. Huang, B. Deng, A. Schiftner, M. Kilian, L. Guibas, and J. Wallner. Geodesic patterns. ACM Trans. Graphics 29/4 (2010), #43,1-10, Proc. SIGGRAPH. [doi].
• C. Müller. Conformal hexagonal meshes. Geom. Dedicata 154 (2011), 27-46. [doi].
• A. Bobenko, H. Pottmann, and J. Wallner. A curvature theory for discrete surfaces based on mesh parallelity. Math. Annalen 348 (2010), 1-24. [Zbl], [MR], [doi], [arxiv:0901.4620].
• A. Schiftner, M. Höbinger, J. Wallner, and H. Pottmann. Packing circles and spheres on surfaces. ACM Trans. Graphics 28/5 (2009), #139,1-8, Proc. SIGGRAPH Asia. [doi].
• H. Pottmann, P. Grohs, and N. J. Mitra. Laguerre minimal surfaces, isotropic geometry and linear elasticity. Adv. Comp. Math. 31 (2009), 391-419. [MR].
• O. Karpenkov. Necessary flexibility conditions of semidiscrete surfaces. Geometry Preprint 2010/06, TU Graz, April 2010. [arxiv:1004.2420v1].
• O. Karpenkov. On the flexibility of Kokotsakis meshes. Geom. Dedicata 147 (2010), 15-28. [doi], [arxiv:0812.3050].
• C. Müller and J. Wallner. Oriented mixed area and discrete minimal surfaces. Discrete Comput. Geom. 43 (2010), 303-320. [Zbl], [MR], [doi].
• J. Wallner. Semidiscrete surface representations. In A. Bobenko et al., editors, Discrete Differential Geometry, Oberwolfach Reports. 2009. Abstracts from the workshop held January 12--17, 2009. [doi].
• H. Pottmann, P. Grohs, and B. Blaschitz. Edge offset meshes in Laguerre geometry. Adv. Comp. Math. 33 (2010), 45-73. [MR], [doi].
• H. Pottmann, A. Schiftner, and J. Wallner. Geometry of architectural freeform structures. Int. Math. Nachr. 209 (2008), 15-28.
• H. Pottmann, J. Wallner, Q. Huang, and Y.-L. Yang. Integral invariants for robust geometry processing. Comput. Aided Geom. Design 26 (2009), 37-60. [Zbl], [MR], [doi].
• H. Pottmann, A. Schiftner, P. Bo, H. Schmiedhofer, W. Wang, N. Baldassini, and J. Wallner. Freeform surfaces from single curved panels. ACM Trans. Graphics 27/3 (2008), #76,1-10, Proc. SIGGRAPH. [doi].
• H. Pottmann, Y. Liu, J. Wallner, A. Bobenko, and W. Wang. Geometry of multi-layer freeform structures for architecture. ACM Trans. Graphics 26/3 (2007), #65,1-11, Proc. SIGGRAPH. [doi].
• J. Wallner and H. Pottmann. Infinitesimally flexible meshes and discrete minimal surfaces. Monatshefte Math. 153 (2008), 347-365. [Zbl], [MR], [doi].
• H. Pottmann, S. Brell-Cokcan, and J. Wallner. Discrete surfaces for architectural design. In P. Chenin, T. Lyche, and L. L. Schumaker, editors, Curves and Surface Design: Avignon 2006, pages 213-234. Nashboro Press, 2007, ISBN 978-0-9728482-7-5. [Zbl], [MR].
• H. Pottmann and J. Wallner. The focal geometry of circular and conical meshes. Adv. Comp. Math 29 (2008), 249-268. [Zbl], [MR], [doi].
• Y. Liu, H. Pottmann, J. Wallner, Y.-L. Yang, and W. Wang. Geometric modeling with conical meshes and developable surfaces. ACM Trans. Graphics 25/3 (2006), 681-689, Proc. SIGGRAPH. [doi].
• W. Wang, J. Wallner, and Y. Liu. An angle criterion for conical mesh vertices. J. Geometry Graphics 11 (2007), 199-208. [Zbl], [MR].
• J. Wallner. Computing quadrilateral and conical meshes. In A. Bobenko et al., editors, Discrete Differential Geometry, volume 3 of Oberwolfach Reports. 2006. Abstracts from the workshop held March 6--10, 2006. [Zbl].
• H. Pottmann, J. Wallner, Y.-L. Yang, Y.-K. Lai, and S.-M. Hu. Principal curvatures from the integral invariant viewpoint. Comput. Aided Geom. Design 24 (2007), 428-442. [MR], [doi].