  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 A-net 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 Pluecker geometry to describe A-nets and hyperboloids.
