Architectural Surfaces and Structures from Circular Arcs

Abstract

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.