On Discrete Constant Mean Curvature Surfaces
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Recently, a curvature theory for polyhedral surfaces has been established that associates with each face a mean curvature value computed from areas and mixed areas of that face and its corresponding Gaussian image face. Therefore, a study of constant mean curvature (cmc) surfaces requires studying pairs of polygons with some constant nonvanishing value of the discrete mean curvature for all faces. We focus on meshes where all faces are planar quadrilaterals or planar hexagons. We show an incidence geometric characterization of a pair of parallel quadrilaterals having a discrete mean curvature value of −1. This characterization yields an integrability condition for a mesh being a Gaussian image mesh of a discrete cmc surface. Thus, we can use these geometric results for the construction of discrete cmc surfaces. In the special case where all faces have a circumcircle, we establish a discrete Weierstrass-type representation for discrete cmc surfaces.
KeywordsDiscrete differential geometry Discrete curvatures Discrete cmc surfaces Discrete Weierstrass representation Oriented mixed area Geometric configurations
This research was supported by the DFG Collaborative Research Center TRR 109, “Discretization in Geometry and Dynamics” through grant I 706-N26 of the Austrian Science Fund (FWF).
- 6.Hertrich-Jeromin, U., Hoffmann, T., Pinkall, U.: A discrete version of the Darboux transform for isothermic surfaces. In: Discrete Integrable Geometry and Physics, Vienna, 1996. Oxford Lecture Ser. Math. Appl., vol. 16, pp. 59–81. Oxford University Press, New York (1999) Google Scholar
- 8.Hoffmann, T.: Discrete cmc surfaces and discrete holomorphic maps. In: Discrete Integrable Geometry and Physics, Vienna, 1996. Oxford Lecture Ser. Math. Appl., vol. 16, pp. 97–112. Oxford University Press, New York (1999) Google Scholar
- 11.Liu, Y., Wang, W.: On vertex offsets of polyhedral surfaces. In: Proc. of Advances in Architectural Geometry, pp. 61–64 (2008). TU Wien Google Scholar
- 16.Pottmann, H., Liu, Y., Wallner, J., Bobenko, A.I., Wang, W.: Geometry of multi-layer freeform structures for architecture. ACM Trans. Graphics 26(3), Article 65, 1–11 (2007) Google Scholar