Three-dimensional structures in building construction and architecture are realized with conflicting goals in mind: engineering considerations and financial constraints easily are at odds with creative aims. It would therefore be very beneficial if optimization and side conditions involving statics and geometry could play a role already in early stages of design, and could be incorporated in design tools in an unobtrusive and interactive way. This paper, which is concerned with a prominent class of structures, is a substantial step towards this goal. We combine the classical work of Maxwell, Michell, and Airy with differential-geometric considerations and obtain a geometric understanding of "optimality" of surface-like lightweight structures. It turns out that total absolute curvature plays an important role. We enable the modeling of structures of minimal weight which in addition have properties relevant for building construction and design, like planar panels, dominance of axial forces over bending, and geometric alignment constraints.

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.

Cold bent glass is a promising and cost-efficient method for realizing doubly curved glass façades. They are produced by attaching planar glass sheets to curved frames and must keep the occurring stress within safe limits. However, it is very challenging to navigate the design space of cold bent glass panels because of the fragility of the material, which impedes the form finding for practically feasible and aesthetically pleasing cold bent glass façades. We propose an interactive, data-driven approach for designing cold bent glass façades that can be seamlessly integrated into a typical architectural design pipeline. Our method allows non-expert users to interactively edit a parametric surface while providing real-time feedback on the deformed shape and maximum stress of cold bent glass panels. The designs are automatically refined to minimize several fairness criteria, while maximal stresses are kept within glass limits. We achieve interactive frame rates by using a differentiable Mixture Density Network trained from more than a million simulations. Given a curved boundary, our regression model is capable of handling multistable configurations and accurately predicting the equilibrium shape of the panel and its corresponding maximal stress. We show that the predictions are highly accurate and validate our results with a physical realization of a cold bent glass surface.

Meshes are omnipresent in geometric computing: as a geometry representation, as a basis for simulations, and for shape optimization.

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