We present a discrete theory for modeling developable surfaces as quadrilateral meshes satisfying simple angle constraints. The basis of our model is a lesser known characterization of developable surfaces as manifolds that can be parameterized through orthogonal geodesics. Our model is simple, local, and, unlike previous works, it does not directly encode the surface rulings. This allows us to model continuous deformations of discrete developable surfaces independently of their decomposition into torsal and planar patches or the surface topology. We prove and experimentally demonstrate strong ties to smooth developable surfaces, including a theorem stating that every sampling of the smooth counterpart satisfies our constraints up to second order. We further present an extension of our model that enables a local definition of discrete isometry. We demonstrate the effectiveness of our discrete model in a developable surface editing system, as well as computation of an isometric interpolation between isometric discrete developable shapes.
Keywords: Discrete developable surfaces, discrete differential geometry, geodesic nets, isometry, mesh editing, shape interpolation, shape modeling
We present a 2×2 Lax representation for discrete circular nets of constant negative Gauß curvature. It is tightly linked to the 4D consistency of the Lax representation of discrete K-nets (in asymptotic line parametrization). The description gives rise to Bäcklund transformations and an associated family. All the members of that family - although no longer circular - can be shown to have constant Gauß curvature as well. Explicit solutions for the Bäcklund transformations of the vacuum (in particular Dini's surfaces and breather solutions) and their respective associated families are given.
Keywords: discrete differential geometry, discrete integrable systems, Bäcklund transformations, multidimensional consistency
A. I. Bobenko and T. Hoffmann. Towards a unified theory of discrete surfaces with constant mean curvature. In A. I. Bobenko, editor, Advances in Discrete Differential Geometry, pages 287-308. Springer, 2016. [ bib ]
We introduce a novel class of s-conical nets and, in particular, study s-conical nets with constant mean curvature. Moreover we give a unified description of nets of various types: circular, conical and s-isothermic. The later turn out to be interpolating between the circular net discretization and the s-conical one.
Tim Hoffmann and Andrew O. Sageman-Furnas. A 2x2 lax representation, associated family, and baecklund transformation for circular k-nets. Discrete & Computational Geometry, 56(2):472-501, 2016. [ bib | DOI | arXiv ]
We propose a discrete surface theory in 3 that unites the most prevalent versions of discrete special parametrizations. Our theory encapsulates a large class of discrete surfaces given by a Lax representation and, in particular, the one-parameter associated families of constant curvature surfaces. Our theory is not restricted to integrable geometries, but extends to a general surface theory.
Tim Hoffmann, Wayne Rossman, Takeshi Sasaki, and Masaaki Yoshida. Discrete flat surfaces and linear weingarten surfaces in hyperbolic 3-space. Transactions of the American Mathematical Society, 364(11):5605-5644, 2012. [ bib | arXiv ]
We define discrete flat surfaces in hyperbolic 3-space H3 from the perspective of discrete integrable systems and prove properties that justify the definition. We show how these surfaces correspond to previously defined discrete constant mean curvature 1 surfaces in H3, and we also describe discrete focal surfaces (discrete caustics) that can be used to define singularities on discrete flat surfaces. Along the way, we also examine discrete linear Weingarten surfaces of Bryant type in H3, and consider an example of a discrete flat surface related to the Airy equation that exhibits swallowtail singularities and a Stokes phenomenon.