Rationalization of Freeform Surfaces

An ever broader availability of freeform designs together with an increasing demand for product customization has lead to a rising interest in efficient physical realization of such designs, the trend toward personal fabrication. Not only large-scale architectural applications are (becoming increasingly) popular but also different consumer-level rapid-prototyping applications, including toy and 3D puzzle creation. In this work we present a method for do-it-yourself reproduction of freeform designs without the typical limitation of state-of-the-art approaches requiring manufacturing custom parts using semi-professional laser cutters or 3d printers. Our idea is based on a popular mathematical modeling system (Zometool) commonly used for modeling higher dimensional polyhedra and symmetric structures such as molecules and crystal lattices. The proposed method extends the scope of Zometool modeling to freeform, disktopology surfaces. While being an efficient construction system on the one hand (consisting only of a single node type and 9 different edge types), this inherent discreteness of the Zometool system, on the other hand gives rise to a hard approximation problem. We base our method on a marching front approach, where elements are not added in a greedy sense, but rather whole regions on the front are filled optimally, using a set of problem specific heuristics to keep complexity under control.