Institute of Geometry

This research area has been explored within the framework of the the National Research Network Industrial Geometry, funded by FWF, the SFB-Transregio Programme Discretization in Geometry and Dynamics which is funded by DFG and in part by FWF, and the Doctoral Programme Discrete Mathematics funded byFWF and installed at TU Graz, KFU Graz and MU Leoben.

Computational Differential Geometry means methods of both numerical and discrete mathematics with the purpose of investigating and modeling curves and surfaces. Themes of this research project are 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. A recent topic is isometric mappings, including curved folding. The wealth of interesting geometry connected to applications in, say, architecture has come to the attention of mathematicians only a short while ago, and presumably only a small part of it has been investigated.

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

• V. Ceballos Inza, F. Rist, J. Wallner, and H. Pottmann. Developable quad meshes and contact element nets. ACM Trans. Graph. 42/6 (2023), 183:1-13, Proc. SIGGRAPH Asia. [doi].
• D. Liu, D. Pellis, Y.-C. Chiang, F. Rist, J. Wallner, and H. Pottmann. Deployable strip structures. ACM Trans. Graph. 42/4 (2023), 103:1-16, Proc. SIGGRAPH. [doi].
• C. Jiang, C. Wang, X. Tellier, J. Wallner, and H. Pottmann. Planar panels and planar supporting beams in architectural structures. ACM Trans. Graph. 42/2 (2023), 19:1-17. [doi].
• C. Jiang, F. Rist, H. Wang, J. Wallner, and H. Pottmann. Shape-morphing mechanical metamaterials. Computer-Aided Design 143 (2022), 103146. [doi].
• C. Jiang, H. Wang, V. Ceballos Inza, F. Dellinger, F. Rist, J. Wallner, and H. Pottmann. Using isometries for computational design and fabrication. ACM Trans. Graph. 40/4 (2021), 42:1-12, Proc. SIGGRAPH. [doi].
• L. Alese. Closing curves by rearranging arcs. Colloq. Math. 169 (2022), 197-209. [doi], [arxiv: 2002.05422].
• L. Alese. Propagation of curved folding: The folded annulus with multiple creases exists. Beitr. Algebra Geom. 63 (2022), 19-43. [doi], [arxiv: 2008.02660].
• L. Alese. Affine subspaces of curvature functions from closed planar curves. Archiv Math. 76/2 (2021), 70:1-14. [doi], [arxiv: 2006.09678].
• C. Jiang, F. Rist, H. Pottmann, and J. Wallner. Freeform quad-based kirigami. ACM Trans. Graphics 39/6 (2020), 209:1-11, Proc. SIGGRAPH Asia. [doi].
• J. Wallner. Discrete parametric surfaces. In K. Crane et al., editors, Discrete Differential Geometry, volume 76 of Proc. Symposia Appl. Math, pages 19-39. American Math. Soc., 2020. [doi].
• C. Jiang, C. Wang, F. Rist, J. Wallner, and H. Pottmann. Quad-mesh based isometric mappings and developable surfaces. ACM Trans. Graphics 39/4 (2020), 128:1-13, Proc. SIGGRAPH. [doi].
• D. Pellis, M. Kilian, F. Dellinger, J. Wallner, and H. Pottmann. Visual smoothness of polyhedral surfaces. ACM Trans. Graphics 38/4 (2019), 31:1-31:11, Proc. SIGGRAPH. [doi].
• C. Jiang, K. Mundilova, F. Rist, J. Wallner, and H. Pottmann. Curve-pleated structures. ACM Trans. Graphics 38/6 (2019), 169:1-13, Proc. SIGGRAPH Asia. [doi].
• D. Pellis, M. Kilian, J. Wallner, and H. Pottmann. Material-minimizing forms and structures. ACM Trans. Graphics 36/6 (2017), article 173, Proc. SIGGRAPH Asia. [doi].
• H. Pottmann and J. Wallner. Freeform architecture and discrete differential geometry. In W. Kropatsch, N. Artner, and I. Janusch, editors, Discrete Geometry for Computer Imagery - DGCI 2017, volume 10502 of Lecture Notes in Computer Science, pages 3-8. Springer, 2017, ISBN 978-3-319-66271-8. [doi].
• H. Pottmann and J. Wallner. Geometry and freeform architecture. In W. König, editor, Mathematics and Society, pages 131-151. European Math. Soc., 2016. [MR], [doi].
• C. Jiang, F. Günther, J. Wallner, and H. Pottmann. Measuring and controlling fairness of triangulations. In Advances in Architectural Geometry 2016, pages 24-39. vdf Hochschulverlag, Zürich, 2016. [doi].
• C. Tang, M. Kilian, P. Bo, J. Wallner, and H. Pottmann. Analysis and design of curved support structures. In Advances in Architectural Geometry 2016, pages 8-23. vdf Hochschulverlag, Zürich, 2016. [doi].
• W. Carl. On semidiscrete constant mean curvature surfaces and their associated families. Monatsh. Math. 182 (2017), 537-563. [doi].
• W. Carl and J. Wallner. Variational Laplacians for semidiscrete surfaces. Adv. Comp. Math 42 (2016), 1491-1509. [MR], [doi].
• C. Tang, P. Bo, J. Wallner, and H. Pottmann. Interactive design of developable surfaces. ACM Trans. Graphics 35/2 (2016), 12:1-12. [doi].
• O. Karpenkov. Finite and infinitesimal flexibility of semidiscrete surfaces. Arnold Math. J. 1/4 (2015), 403-444. [doi].
• W. Carl. A Laplace operator on semi-discrete surfaces. Foundations Comp. Math. 16 (2016), 1115-1150. [MR], [doi].
• X. Sun, C. Jiang, J. Wallner, and H. Pottmann. Vertex normals and face curvatures of triangle meshes. In A. I. Bobenko, editor, Advances in Discrete Differential Geometry, pages 267-286. Springer, 2016. [MR], [doi].
• C. Jiang, J. Wang, J. Wallner, and H. Pottmann. Freeform honeycomb structures. Computer Graphics Forum 33/5 (2014), 185-194, Proc. Symp. Geom. Processing. [doi].
• C. Jiang, J. Wang, P. Bompas, H. Pottmann, and J. Wallner. Freeform shading and lighting systems from planar quads. In C. Gengnagel, A. Kilian, J. Nembrini, and F. Scheurer, editors, Rethinking Prototyping - Proc. Design Modelling Symposium Berlin 2013, pages 335-346. epubli GmbH, Berlin, 2013, ISBN 978-3-8442-6845-4.
• J. Wang, C. Jiang, P. Bompas, J. Wallner, and H. Pottmann. Discrete line congruences for shading and lighting. Computer Graphics Forum 32/5 (2013), 53-62, Proc. Symposium Geometry Processing. [doi].
• C. Müller and J. Wallner. Semi-discrete isothermic surfaces. Results Math. 63 (2013), 1395-1407. [Zbl], [MR], [doi].
• O. Karpenkov and J. Wallner. On offsets and curvatures for discrete and semidiscrete surfaces. Beitr. Algebra Geom. 55 (2014), 207-228. [Zbl], [MR], [doi].
• E. Vouga, M. Höbinger, J. Wallner, and H. Pottmann. Design of self-supporting surfaces. ACM Trans. Graphics 31/4 (2012), 87:1-11, Proc. SIGGRAPH. [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. [doi].
• 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. 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].