Isothermic Patches and Property-Preserving Transformations in Shape Generation of Gridshells

This paper investigates the generation and utilization of Isothermic Surfaces (IS), which are distinguished by their unique geometric properties of admitting conformal and conjugate patches, or CC-patches. These properties facilitate planarity of faces, orthogonality of joints, and square-like proportionality of sub-divisions, among others. Our pre-rationalization approach leverages CC-patches across six surface types to develop a design tool that enables the generation of a wide range of variations, i.e. design possibilities. More precisely, reparemetrizations as well as transformations—including Christoffel, Bäcklund-Darboux-Ribaucour (BDR), Goursat and Möbius—are applied to generate and transform the different IS types. While each transformation preserves the CC-patch property, it can alter other geometric properties of the surface. Thus, allowing the designer to choose transformations with more architectural shape variety, while maintaining the mentioned fabrication advantages of CC-patches.