Discrete Surfaces with Applications in Architectural Design

In the last decade free-form geometry has gained increasing popularity in architecture, civil engineering and industrial design. The close relation between shape and fabrication poses new challenges and requires more sophistication from the underlying geometry. The use of laminated safety glass and other materials which are used for coverings of roofing structures requires bending and deformation which often turns out to be expensive, quite complex or even impossible. Prominent companies like Gehry Partners and Schlaich Bergermann and Partners prefer to use planar quadrilateral meshes instead of triangular meshes. Moreover, since layer composition constructions are frequently used in architecture, the planarity of the faces of a mesh has to be guaranteed for all the layers of the component. These practical requirements lead to discrete surface representations with planar quadrilateral meshes. Modeling of these discrete surfaces as well as the approximation problems of free-form geometry by these meshes are important issues. The layer composition constructions require planarity of the facets not only for the base surface but the same type of meshes shall be used for the representation of the offsets as well. The mathematical background of this research proposal is the combination of discrete differential geometry, applied sphere geometry and geometric optimization. It turns out that principal meshes, i.e., the discrete analogues of networks of principal curvature lines, play an important role to solve the present problems. Although discrete differential geometry is an active research area within mathematics, available studies of principal meshes are quite incomplete. The preparations of this proposal have shown that there are two different approaches to define principal meshes, one in the sense of Möbius-geometry, and the other in the sense of Laguerre-geometry. The first method has been partially investigated, while the second one is totally unexplored so far. This new Laguerre-geometric definition of principal meshes is preferable regarding applications to layer composition constructions, since a discrete principal mesh in the base geometry directly implies a discrete principal mesh in the offsets. Moreover these networks share nice aesthetic and important static properties. The main topics of this research project will be the investigation of quadrilateral meshes with planar faces, in particular, the discrete analogues of the network of principal curvature lines. We will develop algorithms for the approximation of surfaces with planar quadrilateral meshes. Particular emphasis is on the design of such meshes in combination with non-linear subdivision algorithms. As a special case this leads to a new, intuitive and simple method for interactive modeling of developable surfaces. The latter also play a special role in the design process of free-form surfaces in architecture. Besides the aesthetic appearance, also aspects of statics and fabrication will be considered within the shape generating optimization process.

In the last decade free-form geometry has gained increasing popularity in architecture, civil engineering and industrial design. The close relation between shape and fabrication poses new challenges and requires more sophistication from the underlying geometry. The use of laminated safety glass and other materials which are used for coverings of roofing structures requires bending and deformation which often turns out to be expensive, quite complex or even impossible. Prominent companies like Gehry Partners and Schlaich Bergermann and Partners prefer to use planar quadrilateral meshes instead of triangular meshes. Moreover, since layer composition constructions are frequently used in architecture, the planarity of the faces of a mesh has to be guaranteed for all the layers of the structure. These practical requirements lead to discrete surface representations with planar quadrilateral meshes. Modeling of these discrete surfaces as well as the approximation problems of free-form geometry by these meshes are important issues. Multi-layer constructions require planarity of the facets not only for the base surface but the same type of meshes shall be used for the representation of the offsets as well. The mathematical background of this research project is the combination of discrete differential geometry and numerical optimization. It turned out that important practical requirement can be nicely translated into the framework of discrete differential geometry. This contributed to the solution of some main problems in this area. The development of corresponding algorithms is based on nonlinear optimization, which succeeded mainly due to the geometric insight and an according proper initialization. The main results of the project from the architectural perspective are new methods and algorithms for the design of quadrilateral meshes with planar faces, of supporting structures with optimized nodes and the construction of multilayer structures. The main mathematical results have been motivated by the architectural requirements and concern a new curvature theory of polyhedral surfaces and new types of meshes with offset properties and remarkable relations to differential sphere geometry.