Abstract
In the category of semidiscrete surfaces with one discrete and one smooth parameter we discuss the asymptotic parametrizations, their Lelieuvre vector fields, and especially the case of constant negative Gaussian curvature. In many aspects these considerations are analogous to the well known purely smooth and purely discrete cases, while in other aspects the semidiscrete case exhibits a different behaviour. One particular example is the derived T-surface, the possibility to define Gaussian curvature via the Lelieuvre normal vector field, and the use of the T-surface’s regression curves in the proof that constant Gaussian curvature is characterized by the Chebyshev property. We further identify an integral of curvatures which satisfies a semidiscrete Hirota equation.
Similar content being viewed by others
References
Bianchi L.: Lezioni di Geometria Differenziale. Spoerri, Pisa (1902)
Blaschke W.: Vorlesungen über Differentialgeometrie. Springer, Berlin (1923)
Bobenko A., Matthes D., Suris Y.: Discrete and smooth orthogonal systems: C ∞-approximation. Int. Math. Res. Not. 45, 2415–2459 (2003)
Bobenko A., Matthes D., Suris Y.: Nonlinear hyperbolic equations in surface theory: integrable discretizations and approximation results. St. Petersburg Math. J. 17, 39–61 (2005)
Bobenko A., Pinkall U.: Discrete surfaces with constant negative Gaussian curvature and the Hirota equation. J. Differ. Geom. 43, 527–611 (1996)
Bobenko A., Suris Y.: On organizing principles of discrete differential geometry, geometry of spheres. Russ. Math. Surv. 62, 1–43 (2007)
Bobenko A., Suris Y.: Discrete differential geometry: integrable Structure, vol. 98. American Mathematical Society, Providence (2008)
Eisenhart L.P.: A Treatise on the Differential Geometry of Curves and Surfaces. Dover, New York (1960)
Liu Y., Pottmann H., Wallner J., Yang Y.-L., Wang W.: Geometric modeling with conical meshes and developable surfaces. ACM Trans. Graph. 25, 681–689 (2006)
Orfanidis S.J.: Discrete sine-Gordon equations. Phys. Rev. D 18, 3822–3827 (1978)
Orfanidis S.J.: Sine-Gordon equation and nonlinear σ model on a lattice. Phys. Rev. D 18, 3828–3832 (1978)
Pottmann, H., Schiftner, A., Bo, P., Schmiedhofer, H., Wang, W., Baldassini, N., Wallner, J.: Freeform surfaces from single curved panels. ACM Trans. Graph. 27, article #76 (2008)
Pottmann H., Schiftner A., Wallner J.: Geometry of architectural freeform structures. Int. Math. Nachr. 209, 15–28 (2008)
Pottmann H., Wallner J.: The focal geometry of circular and conical meshes. Adv. Comp. Math. 29, 249–268 (2008)
Sauer R.: Parallelogrammgitter als Modelle pseudosphärischer Flächen. Math. Z. 52, 611–622 (1950)
Sauer R.: Differenzengeometrie. Springer, Berlin (1970)
Wunderlich W.: Zur Differenzengeometrie der Flächen konstanter negativer Krümmung. Sitz. Öst. Ak. Wiss. 160, 41–77 (1951)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Wallner, J. On the semidiscrete differential geometry of A-surfaces and K-surfaces. J. Geom. 103, 161–176 (2012). https://doi.org/10.1007/s00022-012-0108-4
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00022-012-0108-4