Variational Tangent Plane Intersection for Planar Polygonal Meshing
Several theoretical and practical geometry applications are based on polygon meshes with planar faces. The planar panelization of freeform surfaces is a prominent example from the field of architectural geometry. One approach to obtain a certain kind of such meshes is by intersection of suitably distributed tangent planes. Unfortunately, this simple tangent plane intersection (TPI) idea has a number of limitations. It is restricted to the generation of hexagon-dominant meshes: as vertices are in general defined by three intersecting planes, the resulting meshes are basically duals of triangle meshes. Furthermore, the explicit computation of intersection points requires dedicated handling of special cases and degenerate constellations to achieve robustness on freeform surfaces. Another limitation is the small number of degrees of freedom for incorporating design parameters.
Using a variational re-formulation, we equip the concept of TPI with additional degrees of freedom and present a robust, unified approach for creating polygonal structures with planar faces that is readily able to integrate various objectives and constraints needed in different applications scenarios. We exemplarily demonstrate the abilities of our approach on three common problems in geometry processing.
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- BOBENKO, A., AND SURIS, Y. 2008. Discrete Differential Geometry: Integrable Structure. Graduate Studies in Mathematics. American Mathematical Society.Google Scholar
- BOMMES, D., ZIMMER, H., AND KOBBELT, L. 2010. Practical Mixed-Integer Optimization for Geometry Processing. In Curves and Surfaces, Springer, J.-D. Boissonnat, P. Chenin, A. Cohen, C. Gout, T. Lyche, M.-L. Mazure, and L. L. Schumaker, Eds., vol. 6920 of Lecture Notes in Computer Science, 193–206.Google Scholar
- CUTLER, B., AND WHITING, E. 2007. Constrained planar remeshing for architecture. In Proceedings of Graphics Interface 2007, ACM, GI’ 07, 11–18.Google Scholar
- MÖBIUS, J., AND KOBBELT, L. 2010. OpenFlipper: An Open Source Geometry Processing and Rendering Framework. In Curves and Surfaces, Springer, J.-D. Boissonnat, P. Chenin, A. Cohen, C. Gout, T. Lyche, M.-L. Mazure, and L. L. Schumaker, Eds., vol. 6920 of Lecture Notes in Computer Science, 488–500.Google Scholar
- POTTMANN, H., BRELL-COKCAN, S., AND WALLNER, J. 2007. Discrete surfaces for architectural design. In Curves and Surface Design: Avignon 2006, P. Chenin, T. Lyche, and L. L. Schumaker, Eds. Nashboro Press, 213–234.Google Scholar
- POTTMANN, H., LIU, Y., WALLNER, J., BOBENKO, A., AND WANG, W. 2007. Geometry of multi-layer freeform structures for architecture. ACM Trans. Graph. 26, 3, 65:1–65:11.Google Scholar
- TRAUTZ, M., AND HERKRATH, R. 2009. The application of folded plate principles on spatial structures with regular, irregular and free-form geometries. In Proc. IASS, 1019–1031.Google Scholar
- TROCHE, C. 2008. Planar hexagonal meshes by tangent plane intersection. In Advances in Architectural Geometry.Google Scholar
- WANG, W., LIU, Y., YAN, D., CHAN, B., LING, R., AND SUN, F. 2008. Hexagonal meshes with planar faces. Tech. Rep. TR-2008-13, Department of Computer Science, The University of Hong Kong.Google Scholar
- YANG, Y.-L., YANG, Y.-J., POTTMANN, H., AND MITRA, N. J. 2011. Shape space exploration of constrained meshes. ACM Trans. Graph. 30, 6, 124:1–124:12.Google Scholar