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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…
We discuss discretization of Koenigs nets (conjugate nets with equal Laplace invariants) and of isothermic surfaces. Our discretization is based on the notion of dual quadrilaterals: two planar quadrilaterals are called dual, if their…
We study local and global approximations of smooth nets of curvature lines and smooth conjugate nets by respective discrete nets (circular nets and planar quadrilateral nets) with infinitesimal quads. It is shown that choosing the points of…
Cyclidic nets are introduced as discrete analogs of curvature line parametrized surfaces and orthogonal coordinate systems. A 2-dimensional cyclidic net is a piecewise smooth $C^1$-surface built from surface patches of Dupin cyclides, each…
Supercyclides are surfaces with a characteristic conjugate parametrization consisting of two families of conics. Patches of supercyclides can be adapted to a Q-net (a discrete quadrilateral net with planar faces) such that neighboring…
Discrete conjugate systems are quadrilateral nets with all planar faces. Discrete orthogonal systems are defined by the additional property of all faces being concircular. Their geometric properties allow one to consider them as proper…
In this paper we are interested in defining affine structures on discrete quadrangular surfaces of the affine three-space. We introduce, in a constructive way, two classes of such surfaces, called respectively indefinite and definite…
We study discrete curvatures computed from nets of curvature lines on a given smooth surface, and prove their uniform convergence to smooth principal curvatures. We provide explicit error bounds, with constants depending only on properties…
This paper studies the discrete differential geometry of the checkerboard pattern inscribed in a quadrilateral net by connecting edge midpoints. It turns out to be a versatile tool which allows us to consistently define principal nets,…
Two-dimensional affine A-nets in 3-space are quadrilateral meshes that discretize surfaces parametrized along asymptotic lines. The characterizing property of A-nets is planarity of vertex stars, so for generic A-nets the elementary…
The focus is on circular nets with one or two families of spherical parameter lines, which are treated in M\"obius geometry. These circular nets provide a discretisation of surfaces with one or two families of spherical curvature lines. The…
Asymptotic net is an important concept in discrete differential geometry. In this paper, we show that we can associate affine discrete geometric concepts to an arbitrary non-degenerate asymptotic net. These concepts include discrete affine…
We consider in this paper discrete improper affine spheres based on asymptotic nets. In this context, we distinguish the discrete edges and vertices that must be considered singular. The singular edges can be considered as discrete cuspidal…
In this paper, we study some relationships existing between some particular mathematical structures: discrete surfaces coming from discrete topology and mathematical morphology, poset-based connected manifolds coming from discrete topology,…
We present a large family of Spin(p,q)-valued discrete spectral problems. The associated discrete nets generated by the so called Sym-Tafel formula are circular nets (i.e., all elementary quadrilaterals are inscribed into circles). These…
A `discrete differential manifold' we call a countable set together with an algebraic differential calculus on it. This structure has already been explored in previous work and provides us with a convenient framework for the formulation of…
Discrete Koenigs nets are a special class of discrete surfaces that play a fundamental role in discrete differential geometry, in particular in the study of discrete isothermic and minimal surfaces. Recently, it was shown by Bobenko and…
Multilayer networks represent systems in which there are several topological levels each one representing one kind of interaction or interdependency between the systems' elements. These networks have attracted a lot of attention recently…
Conjugate line parametrizations of surfaces were first discretized almost a century ago as quad meshes with planar faces. With the recent development of discrete differential geometry, two discretizations of principal curvature line…
Principal binets are a discretization of curvature line parametrized surfaces defined on the vertices and faces of the square lattice $\Z^2$. They generalize the previously established discretizations given by circular nets, conical nets,…