Industrial Geometry

This project started only in the second funding period of this National Research Network.

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.

1. 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].
2. C. Müller and J. Wallner. Oriented mixed area and discrete minimal surfaces. Geometry Preprint 2008/01, TU Graz, February 2008.
3. H. Pottmann, J. Wallner, Q. Huang, and Y.-L. Yang. Integral invariants for robust geometry processing. Comput. Aided Geom. Design (2008), to appear. [doi].
4. 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].
5. J. Wallner and H. Pottmann. Infinitesimally flexible meshes and discrete minimal surfaces. Monatshefte Math. 153/347-365 (2008). [MR], [doi].
6. 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. [MR].
7. H. Pottmann and J. Wallner. The focal geometry of circular and conical meshes. Adv. Comp. Math (2008), to appear. [doi].
8. 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].
9. W. Wang, J. Wallner, and Y. Liu. An angle criterion for conical mesh vertices. J. Geometry Graphics 11 (2007), 199-208.
10. J. Wallner. Computing quadrilateral and conical meshes. In A. Bobenko, R. K. J. Sullivan, and G. Ziegler, editors, Discrete differential Geometry, volume 3 of Oberwolfach Reports. 2006. Abstracts from the workshop held March 6--10, 2006. [Zbl].
11. 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].