In recent decades, the popularity of freeform shapes in contemporary architecture poses new challenges to digital design. One of them is the process of rationalization, i.e. to make freeform skins or structures affordable to manufacture, which draws the most attention from geometry researchers. In this thesis, we aim to realize this process with simple geometric primitives, circular arcs. We investigate architectural surfaces and structures consisting of circular arcs. Our focus is lying on how to employ them nicely and repetitively in architectural design, in order to decrease the cost in manufacturing. Firstly, we study Darboux cyclides, which are algebraic surfaces of order ≤ 4. We provide a computational tool to identify all families of circles on a given cyclide based on the spherical model of M ̈obius geometry. Practical ways to design cyclide patches that pass through certain inputs are presented. In particular, certain triples of circle families on Darboux cyclides may be suitably arranged as 3-webs. We provide a complete classification of all possible 3-webs of circles on Darboux cyclides. We then investigate the circular arc snakes, which are smooth sequences of circu- lar arcs. We evolve the snakes such that their curvature, as a function of arc length, remains unchanged. The evolution of snakes is utilized to approximate given surfaces by circular arcs or to generated freeform shapes, and it is realized by a 2-step pro- cess. More interestingly, certain 6-arc snake with boundary constraints can produce a smooth self motion, which can be employed to build flexible structures. Another challenging topic is approximating smooth freeform skins with simple panels. We contribute to this problem area by approximating a negatively-curved 5 surface with a smooth union of rational bilinear patches. We provide a proof for vertex consistency of hyperbolic nets using the CAGD approach of the rational B ́ezier form. Moreover, we use Darboux transformations for the generation of smooth sur- faces composed of Darboux cyclide patches. In this way we not only eliminate the restriction to surfaces with negative Gaussian curvature, but, also obtain surfaces consisting of circular arcs.

Discrete Differential Geometry (DDG) concerns discrete counterparts of notions and methods in differential geometry. This thesis deals with a core subject in DDG, discrete curvature theories on various types of polyhedral surfaces that are practically important for free-form architecture, sunlight-redirecting shading systems, and face recognition. Modeled as polyhedral surfaces, the shapes of free-form structures may have to satisfy different geometric or physical constraints. We study a combination of geometry and physics – the discrete surfaces that can stand on their own, as well as having proper shapes for the manufacture. These proper shapes, known as circular and conical meshes, are closely related to discrete principal curvatures. We study curvature theories that make such surfaces possible. Shading systems of freeform building skins are new types of energy-saving structures that can re-direct the sunlight. From these systems, discrete line congruences across polyhedral surfaces can be abstracted. We develop a new curvature theory for polyhedral surfaces equipped with normal congruences – a particular type of congruences defined by linear interpolation of vertex normals. The main results are a discussion of various definitions of normality, a detailed study of the geometry of such congruences, and a concept of curvatures and shape operators associated with the faces of a triangle mesh. These curvatures are compatible with both normal congruences and the Steiner formula. In addition to architecture, we consider the role of discrete curvatures in face recognition. We use geometric measure theory to introduce the notion of asymptotic cones associated with a singular subspace of a Riemannian manifold, which is an extension of the classical notion of asymptotic directions. We get a simple expression of these cones for polyhedral surfaces, as well as convergence and approximation theorems. We use the asymptotic cones as facial descriptors and demonstrate the practicability and accuracy of their applications in face recognition.

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 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.