Isotropic Geometry and Applications in Geometric Computing

This thesis addresses challenges in computational design and fabrication, particularly in the geometry of architectural gridshell structures and their approximation, by solving nonlinear optimization problems. It uses isotropic geometry, a simplified version of Euclidean geometry, to improve initial guesses for optimization, e.g., by using the semi-definite product $\langle p,q \rangle_i=p_1q_1+p_2q_2$ instead of the inner product in Euclidean geometry $\langle p,q \rangle=p_1q_1+p_2q_2 + p_3q_3$. Isotropic geometry proves helpful in solving Euclidean optimization problems by smoothly transitioning from isotropic to Euclidean geometry during optimization. Applications are illustrated through various architectural structures, particularly those constructed from flat, straight lamellas from thin metal or wood that are bent towards gridshells. These lamellas are arranged in two, three, or four families, forming what is known as a 2-web, 3-web, or 4-web, respectively. In the final, bent structure, lamellas are either tangential, following geodesics (labeled as G), or orthogonal, following asymptotic curves (labeled as A), on the reference surface. We explore 3-webs consisting of three families of geodesics (GGG-webs). We also study 3-webs formed by two families of asymptotic curves and one family of geodesics (AAG-webs). In addition, we consider 4-webs consisting of two families of geodesics and two families of asymptotic curves (AGAG-webs). Motivated by asymptotic gridshells with a constant node angle, this thesis also contributes to the theory of surfaces with constant principal curvature ratios in Euclidean and isotropic geometries. We characterize rotational, channel, ruled, helical, and translational surfaces under certain technical constraints (with the last two done only in isotropic geometry). Furthermore, we prove that any minimal surface with analytic boundary and without flat points is contained in a family of surfaces with a constant ratio of principal curvatures in isotropic and Euclidean $3$-space, which gives a partial solution to the Plateau problem for such surfaces.