No abstract

A Darboux transformation for polarized space curves is introduced and its properties are studied, in particular, Bianchi permutability. Semi-discrete isothermic surfaces are described as sequences of Darboux transforms of polarized curves in the conformal n-sphere and their transformation theory is studied. Semi-discrete surfaces of constant mean curvature are studied as an application of the transformation theory.

Abstract. Recently a curvature theory for polyhedral surfaces has been established which associates with each face a mean curvature value computed from areas and mixed areas of that face and its corresponding Gauss image face. Therefore a study of minimal surfaces requires studying pairs of polygons with vanishing mixed area. We show that the mixed area of two edgewise parallel polygons equals the mixed area of a derived polygon pair which has only the half number of vertices. Thus we are able to recursively characterize vanishing mixed area for hexagons and other n-gons in an incidence-geometric way. We use these geometric results for the construction of discrete minimal surfaces and a study of equilibrium forces in their edges, especially those with the combinatorics of a hexagonal mesh.

Figure 1: A panorama of the capabilities of our framework. Deformation of a circular mesh (left). Metric-conformal deformation and interpolation in 3D (center), and an intersection-angle preserving deformation of a planar mesh (right). AbstractWe establish a framework to design triangular and circular polygonal meshes by using face-based compatible Möbius transformations. Embracing the viewpoint of surfaces from circles, we characterize discrete conformality for such meshes, in which the invariants are circles, cross-ratios, and mutual intersection angles. Such transformations are important in practice for editing meshes without distortions or loss of details. In addition, they are of substantial theoretical interest in discrete differential geometry. Our framework allows for handle-based deformations, and interpolation between given meshes with controlled conformal error.

Geodesic parallel coordinates are orthogonal nets on surfaces where one of the two families of parameter lines are geodesic curves. We describe a discrete version of these special surface parameterizations and show that they are very useful for specific applications, most of which are related to the design and fabrication of surfaces in architecture. With the new discrete surface model, it is easy to control strip widths between neighboring geodesics. This facilitates tasks such as cladding a surface with strips of originally straight flat material or designing geodesic gridshells and timber rib shells. It is also possible to model nearly developable surfaces. These are characterized by geodesic strips with almost constant strip widths and are used for generating shapes that can be manufactured from materials which allow for some stretching or shrinking like felt, leather, or thin wooden boards. Most importantly, we show how to constrain the strip width parameters to model a class of intrinsically symmetric surfaces. These surfaces are isometric to surfaces of revolution and can be covered with doubly-curved panels that are produced with only a few molds when working with flexible materials like metal sheets.