Interactive design of discrete Voss nets and simulation of their rigid foldings

Abstract

Voss nets are surface parametrizations whose parameter lines follow a conjugate network of geodesics. Their discrete counterparts, so called V-hedra, are flexible quadrilateral meshes with planar faces such that opposite angles at every vertex are equal; by replacing this equality condition with the closely related constraint that opposite angles are supplementary, we get so-called anti-V-hedra. In this paper, we study the problem of constructing and manipulating (anti-)V-hedra. First, we present a V-hedra generator that constructs a modifiable (anti-)V-hedron in a geometrically exact way, from a set of simple conditions already proposed by Sauer in 1970; our generator can compute and visualize the flexion of the (anti-)V-hedron in real time. Second, we present an algorithm for the design and interactive exploration of V-hedra using a handle-based deformation approach; this tool is capable of simulating the one-parametric isometric deformation of an imperfect V-hedron via a quad soup approach. Moreover, we evaluate the performance and accuracy of our tools by applying the V-hedra generator to constraints obtained by numerical optimization. In particular, we use example surfaces that originate from one-dimensional families of smooth Voss surfaces – each spanned by two isothermal conjugate nets – for which an explicit parametrization is given. This allows us to compare the isometric deformation of the smooth target surface with the rigid folding of the optimized (imperfect) V-hedron.