Geometry of Multi-layer Freeform Structures for Architecture

The geometric challenges in the architectural design of freeform shapes come mainly from the physical realization of beams and nodes in singleand multilayer constructions. The present paper approaches arising geometric problems via the concept of parallel meshes. It presents results which form the basis for computation and optimization. We discuss planar faces, beams of controlled height, geometrically optimal nodes, and multilayer constructions. Beams of constant height are achieved with a new type of meshes which possess planar faces and the edge offset property. It turns out that mesh parallelism is also the main ingredient in a novel discrete theory of relative curvatures of meshes with planar faces which can be nicely integrated into the overall framework. The presented methods are applied to the construction of pentagonal and hexagonal meshes, discrete minimal surfaces, discrete constant mean curvature surfaces and geometric transforms of those. It is the aim of the present paper to show two processing pipelines: One for designing geometrically optimal shapes, and another one from an existing design to a feasible meshing and supporting beam layout. CR Categories: I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling—Geometric algorithms, languages, and systems; I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling—Curve, surface, solid, and object representations