Self-supporting structures are prominent in historical and contemporary architecture due to advantageous structural properties and efficient use of material. Computer graphics research has recently contributed new design tools that allow creating and interactively exploring self-supporting freeform designs. However, the physical construction of such freeform structures remains challenging, even on small scales. Current construction processes require extensive formwork during assembly, which quickly leads to prohibitively high construction costs for realizations on a building scale. This greatly limits the practical impact of the existing freeform design tools. We propose to replace the commonly used dense formwork with a sparse set of temporary chains. Our method enables gradual construction of the masonry model in stable sections and drastically reduces the material requirements and construction costs. We analyze the input using a variational method to find stable sections, and devise a computationally tractable divide-and-conquer strategy for the combinatorial problem of finding an optimal construction sequence. We validate our method on 3D printed models, demonstrate an application to the restoration of historical models, and create designs of recreational, collaborative self-supporting puzzles.

Abstract Mesh editing under constraints is a challenging task with numerous applications in geometric modeling, industrial design, and architectural form finding. Recent methods support constraint‐based exploration of meshes with fixed connectivity, but commonly lack local control. Because constraints are often globally coupled, a local modification by the user can have global effects on the surface, making iterative design exploration and refinement difficult. Simply fixing a local region of interest a priori is problematic, as it is not clear in advance which parts of the mesh need to be modified to obtain an aesthetically pleasing solution that satisfies all constraints. We propose a novel framework for exploring local modifications of constrained meshes. Our solution consists of three steps. First, a user specifies target positions for one or more vertices. Our algorithm computes a sparse set of displacement vectors that satisfies the constraints and yields a smooth deformation. Then we build a linear subspace to allow realtime exploration of local variations that satisfy the constraints approximately. Finally, after interactive exploration, the result is optimized to fully satisfy the set of constraints. We evaluate our framework on meshes where each face is constrained to be planar.

Abstract We introduce a unified optimization framework for geometry processing based on shape constraints. These constraints preserve or prescribe the shape of subsets of the points of a geometric data set, such as polygons, one‐ring cells, volume elements, or feature curves. Our method is based on two key concepts: a shape proximity function and shape projection operators. The proximity function encodes the distance of a desired least‐squares fitted elementary target shape to the corresponding vertices of the 3D model. Projection operators are employed to minimize the proximity function by relocating vertices in a minimal way to match the imposed shape constraints. We demonstrate that this approach leads to a simple, robust, and efficient algorithm that allows implementing a variety of geometry processing applications, simply by combining suitable projection operators. We show examples for computing planar and circular meshes, shape space exploration, mesh quality improvement, shape‐preserving deformation, and conformal parametrization. Our optimization framework provides a systematic way of building new solvers for geometry processing and produces similar or better results than state‐of‐the‐art methods.