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, disk-topology surfaces. While being an efficient construction system on the one hand (consisting only of a single node type and nine 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.

In the context of shape segmentation and retrieval object-wide distributions of measures are needed to accurately evaluate and compare local regions of shapes. Lien et al. [16] proposed two point-wise concavity measures in the context of Approximate Convex Decompositions of polygons measuring the distance from a point to the polygon's convex hull: an accurate Shortest Path-Concavity (SPC) measure and a Straight Line-Concavity (SLC) approximation of the same. While both are practicable on 2D shapes, the exponential costs of SPC in 3D makes it inhibitively expensive for a generalization to meshes [14]. In this paper we propose an efficient and straight forward approximation of the Shortest Path-Concavity measure to 3D meshes. Our approximation is based on discretizing the space between mesh and convex hull, thereby reducing the continuous Shortest Path search to an efficiently solvable graph problem. Our approach works out-of-the-box on complex mesh topologies and requires no complicated handling of genus. Besides presenting a rigorous evaluation of our method on a variety of input meshes, we also define an SPC-based Shape Descriptor and show its superior retrieval and runtime performance compared with the recently presented results on the Convexity Distribution by Lian et al. [12].

Abstract In mechanical engineering and architecture, structural elements with low material consumption and high load‐bearing capabilities are essential for light‐weight and even self‐supporting constructions. This paper deals with so called point‐folding elements – non‐planar, pyramidal panels, usually formed from thin metal sheets, which exploit the increased structural capabilities emerging from folds or creases. Given a triangulated free‐form surface, a corresponding point‐folding structure is a collection of pyramidal elements basing on the triangles. User‐specified or material‐induced geometric constraints often imply that each individual folding element has a different shape, leading to immense fabrication costs. We present a rationalization method for such structures which respects the prescribed aesthetic and production constraints and finds a minimal set of molds for the production process, leading to drastically reduced costs. For each base triangle we compute and parametrize the range of feasible folding elements that satisfy the given constraints within the allowed tolerances. Then we pose the rationalization task as a geometric intersection problem, which we solve so as to maximize the re‐use of mold dies. Major challenges arise from the high precision requirements and the non‐trivial parametrization of the search space. We evaluate our method on a number of practical examples where we achieve rationalization gains of more than 90%.

We present a novel method for quadrangulating a given triangle mesh. After constructing an as smooth as possible symmetric cross field satisfying a sparse set of directional constraints (to capture the geometric structure of the surface), the mesh is cut open in order to enable a low distortion unfolding. Then a seamless globally smooth parametrization is computed whose iso-parameter lines follow the cross field directions. In contrast to previous methods, sparsely distributed directional constraints are sufficient to automatically determine the appropriate number, type and position of singularities in the quadrangulation. Both steps of the algorithm (cross field and parametrization) can be formulated as a mixed-integer problem which we solve very efficiently by an adaptive greedy solver. We show several complex examples where high quality quad meshes are generated in a fully automatic manner.