Discretization of asymptotic line parametrizations using hyperboloid surface patches

Abstract

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.

Architectural geometry

“…However, smoothly joined rational bilinear patches (called hyperbolic nets) are sufficient to approximate arbitrarily simply connected surfaces. This has first been proved by Huhnen-Venedey and Rörig [48] using the Plücker quadric model of line geometry. A CAGD approach based on the rational Bezier representation and algorithms for remeshing are due to Shi et al [95] (see Fig.…”

On Weingarten Transformations of Hyperbolic Nets

“…Previous results. In the following, we give a brief overview of the work [HVR13], which introduced hyperbolic nets as a novel discretization of smooth A-surfaces.…”

“…According to (19), the value ρ 4 is then obtained as ρ 4 = ρ 3 ρ 6 /ρ 1 aã and the resulting cross for the quadrilateral (x 2 , x 3 , x 4 , x 5 ) is internal if and only if ρ 4 > 0, i.e., if and only if the parallel invariant aã is positive. In [HVR13] it was proven that it is possible to equip two adjacent skew quadrilaterals of an A-net with adapted hyperboloid patches that satisfy the C 1 -condition if and only if the quadrilaterals are equi-twisted (cf. Section 3.2).…”

Multi-Nets. Classification of Discrete and Smooth Surfaces with Characteristic Properties on Arbitrary Parameter Rectangles

“…In previous work we have studied different kinds of extensions of discrete nets by suitable surface patches. This has led to the extensions of circular nets by Dupin cyclide patches [4], of discrete asymptotic nets by hyperboloid patches [19], and of discrete conjugate nets by supercyclide patches [5]. All the surface patches used for the extensions, i.e., Dupin cyclide, hyperboloid, and supercyclide surface patches, are the "simplest" non-trivial surfaces contained in the corresponding smooth surface classes that are characterized by the multi-nets (cf.…”