Discretization of asymptotic line parametrizations using hyperboloid patches

TL;DR

This paper introduces hyperbolic nets, a new discretization method for surfaces parametrized along asymptotic lines, by extending affine A-nets with hyperboloid patches under specific combinatorial and geometric conditions.

Contribution

It classifies extendable affine A-nets into hyperbolic nets and describes conditions for their construction, providing a novel piecewise smooth discretization of asymptotic line surfaces.

Findings

Affine A-nets can be extended to hyperbolic nets with even degree vertices and equi-twisted quadrilateral strips.

Hyperbolic nets form a 1-parameter family of C^1-surfaces.

The paper outlines a computational approach for generating hyperbolic nets.

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 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.