Discrete Surfaces of Constant Ratio of Principal Curvatures

Discrete Surfaces of Constant Ratio of Principal Curvatures Mohammed Alhajji The topic of this thesis is motivated by recent developments in Architectural Geometry, namely Eike Schling’s asymptotic gridshells and progress in solutions for paneling freeform facades. An asymptotic gridshell is fabricated from flat straight lamellas of bendable material such as sheet metal. These strips are then arranged in a grid-like spatial structure, such that the lamellas are orthogonal to a reference surface, which however is not materialized. Differential geometry then tells us that the strips must follow asymptotic curves of that reference surface. The actual construction is simplified if angles at nodes are constant. If that angle is a right angle, one gets minimal surfaces as reference surfaces. If the angle is constant, one obtains negatively curved surfaces with a constant ratio of principal curvatures (CRPC surfaces). Their characteristic parameterizations are equi-angular asymptotic parameterizations. We are also interested in the positively curved CRPC surfaces. Due to the relation between curvatures, they have a one-parameter family of curvature elements, which facilitates cost-effective paneling solutions through mold-reuse. Our approach to positively curved CRPCS surfaces is again based on equi-angular characteristic parameterizations. These characteristic parameterizations are conjugate and symmetric with respect to the principal curvature directions. After a review of the required results from classical surface theory, we first present a derivation of rotational CRPC surfaces. By simple geometric considerations one can find their characteristic parameterizations. In this way we add some new insight to this known class of surfaces. However, it turns out to be very hard to come up 5 with explicit results on non-rotational CRPC surfaces. This is in big contrast to the special case of minimal surfaces which are characterized be the constant principal curvature ratio -1. Due to the difficulties in handling smooth CRPC surfaces, we turn to discrete models in form of constrained quad meshes. The discrete models belong to the area of Discrete Differential Geometry. There, one does not discretize equations from the smooth theory, but fundamental concepts of the theory. We introduce the basic structures needed in this context: asymptotic nets, conjugate nets and principal symmetric nets. The latter are a recent development in discrete differential geometry and characterized by spherical vertex stars. This means that a vertex of the quad mesh and its four connected neighbors lie on a sphere. If that sphere degenerates to a plane at all vertices, one has the classical discrete asymptotic parameterization as an A-net. Several ways to discretize the constant intersection angle are presented. The actual computation of discrete CRPC surfaces is performed with numerical optimization with an appropriately regularized Gauss-Newton algorithm for solving a nonlinear least squares problem. Optimization requires initial configurations. Those can come from the known classes of CRPC surfaces such as rotational surfaces of minimal surfaces. The latter case yields some surprising results on negatively curves CRPC surfaces of nontrivial topology. In general, such discrete models can serve as a guiding line for future research on the theoretical side. This is briefly indicated in the final discussion on future research directions.