A self-Airy membrane shell is a special type of shell structure whose shape coincides with the shell’s Airy stress surface. It provides the convenient property that any polyhedral discretization of such a surface will automatically generate a mesh in funicular equilibrium. A self-Airy shell designed for a uniform vertical load would simply have a constant isotropic Gaussian curvature. However, a challenge in implementing a self-Airy shell in architecture is the lack of a design method, especially in designing unreinforced boundaries. Those are singular planar curves, where the two principal curvatures approach 0 and individually. This paper presents methods for designing unreinforced boundaries of self-Airy shells, including both smooth and discrete methods. These methods work for both positively and negatively curved surfaces. The proposed methods work linearly without iteration. The preliminary results show that the seemingly very restrictive conditions admit a variety of non-trivial surfaces.

Straight flat strips of inextensible material can be bent into curved strips aligned with arbitrary space curves. The large shape variety of these so-called rectifying strips makes them candidates for shape modeling, especially in applications such as architecture where simple elements are preferred for the fabrication of complex shapes. In this paper, we provide computational tools for the design of shapes from rectifying strips. They can form various patterns and fulfill constraints which are required for specific applications such as gridshells or shading systems. The methodology is based on discrete models of rectifying strips, a discrete level-set formulation and optimization-based constrained mesh design and editing. We also analyse the geometry at nodes and present remarkable quadrilateral arrangements of rectifying strips with torsion-free nodes.

We solve the task of representing free forms by an arrangement of panels that are manufacturable by precise isometric bending of surfaces made from a small number of molds. In fact we manage to solve the paneling task with surfaces of constant Gaussian curvature alone. This includes the case of developable surfaces which exhibit zero curvature. Our computations are based on an existing discrete model of isometric mappings between surfaces which for this occasion has been refined to obtain higher numerical accuracy. Further topics are interesting connections of the paneling problem with the geometry of Killing vector fields, designing and actuating isometries, curved folding in the double-curved case, and quad meshes with rigid faces that are nevertheless flexible.

The isolines of principal symmetric surface parametrizations run symmetrically to the principal directions. We describe two discrete versions of these special nets/quad meshes which are dual to each other and show their usefulness for various applications in the context of fabrication and architectural design. Our discretization of a principal symmetric mesh comes naturally with a family of spheres, the so-called Meusnier and Mannheim spheres. In our representation of principal symmetric meshes, we have direct control over the radii of theses spheres and the intersection angles of the parameter lines. This facilitates tasks such as generating Weingarten surfaces including constant mean curvature surfaces and minimal surfaces. We illustrate the potential of Weingarten surfaces for paneling doubly curved freeform facades by significantly reducing the number of necessary molds. Moreover, we have direct access to curvature adaptive tool paths for cylindrical CNC milling with circular edges as well as flank milling with rotational cones. Furthermore, the construction of curved support structures from congruent circular strips is easily managed by constant sphere radii. The underlying families of spheres are in a natural way discrete curvature spheres in analogy to smooth Möbius and Laguerre geometry which further leads to a novel discrete curvature theory for principal symmetric meshes.

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.