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 quadrilaterals are skew. We classify the simply connected affine A-nets that can be extended to continuously differentiable surfaces by gluing hyperboloid surface patches into the skew quadrilaterals. The resulting surfaces are called "hyperbolic nets" and are a novel piecewise smooth discretization of surfaces parametrized along asymptotic lines. It turns out that a simply connected affine Anet has to satisfy one combinatorial and one geometric condition to be extendable -all vertices have to be of even degree and all quadrilateral strips have to be "equi-twisted". Furthermore, if an A-net can be extended to a hyperbolic net, then there exists a 1-parameter family of such C 1 -surfaces. It is briefly explained how the generation of hyperbolic nets can be implemented on a computer. The article uses the projective model of Plücker geometry to describe A-nets and hyperboloids.

We investigate the common underlying discrete structures for various smooth and discrete nets. The main idea is to impose the characteristic properties of the nets not only on elementary quadrilaterals but also on larger parameter rectangles. For discrete planar quadrilateral nets, circular nets, Q * -nets and conical nets we obtain a characterization of the corresponding discrete multi-nets. In the limit these discrete nets lead to some classical classes of smooth surfaces. Furthermore, we propose to use the characterized discrete nets as discrete extensions for the nets to obtain structure preserving subdivision schemes.2010 Mathematics Subject Classification. 51A05, 53A20, 65D17 (Primary), 51B10, 51B15 (Secondary).

While a generic smooth Ribaucour sphere congruence admits exactly two envelopes, a discrete R-congruence gives rise to a 2-parameter family of discrete enveloping surfaces. The main purpose of this paper is to gain geometric insights into this ambiguity. In particular, discrete R-congruences that are enveloped by discrete channel surfaces and discrete Legendre maps with one family of spherical curvature lines are discussed.

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We prove that the neighborly cubical polytopes studied by Günter M. Ziegler and the first named author [14] arise as a special case of the neighborly cubical spheres constructed by Babson, Billera and Chan [4]. By relating the two constructions we obtain an explicit description of a non-polytopal neighborly cubical sphere and, further, a new proof of the fact that the cubical equivelar surfaces of McMullen, Schulz and Wills [16] can be embedded into R 3 .