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.

Figure 1: 3D designs composed of planar intersecting pieces fabricated using laser cutting and CNC milling. AbstractWe propose a computational design approach to generate 3D models composed of interlocking planar pieces. We show how intricate 3D forms can be created by sliding the pieces into each other along straight slits, leading to a simple construction that does not require glue, screws, or other means of support. To facilitate the design process, we present an abstraction model that formalizes the main geometric constraints imposed by fabrication and assembly, and incorporates conditions on the rigidity of the resulting structure. We show that the tight coupling of constraints makes manual design highly nontrivial and introduce an optimization method to automate constraint satisfaction based on an analysis of the constraint relation graph. This algorithm ensures that the planar parts can be fabricated and assembled. We demonstrate the versatility of our approach by creating 3D toy models, an architectural design study, and several examples of functional furniture.

Figure 1: Caustic Brain: Our algorithm computes a 3D surface that refracts uniform light to focus on sharp intensity lines that sketch a human brain. The physical prototype shown on the right has been fabricated in transparent acrylic with a CNC milling machine. The photographs illustrate how the caustic image evolves as the acrylic piece is rotated into position (see also Figure 11 and accompanying video). AbstractWe present a new algorithm for computational caustic design. Our algorithm solves for the shape of a transparent object such that the refracted light paints a desired caustic image on a receiver screen. We introduce an optimal transport formulation to establish a correspondence between the input geometry and the unknown target shape. A subsequent 3D optimization based on an adaptive discretization scheme then finds the target surface from the correspondence map. Our approach supports piecewise smooth surfaces and non-bijective mappings, which eliminates a number of shortcomings of previous methods. This leads to a significantly richer space of caustic images, including smooth transitions, singularities of infinite light density, and completely black areas. We demonstrate the effectiveness of our approach with several simulated and fabricated examples.

In architectural design, surface shapes are commonly subject to geometric constraints imposed by material, fabrication or assembly. Rationalization algorithms can convert a freeform design into a form feasible for production, but often require design modifications that might not comply with the design intent. In addition, they only offer limited support for exploring alternative feasible shapes, due to the high complexity of the optimization algorithm.We address these shortcomings and present a computational framework for interactive shape exploration of discrete geometric structures in the context of freeform architectural design. Our method is formulated as a mesh optimization subject to shape constraints. Our formulation can enforce soft constraints and hard constraints at the same time, and handles equality constraints and inequality constraints in a unified way. We propose a novel numerical solver that splits the optimization into a sequence of simple subproblems that can be solved efficiently and accurately.Based on this algorithm, we develop a system that allows the user to explore designs satisfying geometric constraints. Our system offers full control over the exploration process, by providing direct access to the specification of the design space. At the same time, the complexity of the underlying optimization is hidden from the user, who communicates with the system through intuitive interfaces.

Figure 1: Local modifications of a constrained mesh. In this example a glass structure composed of planar quads is locally deformed by exploring a subspace encoding local planar modifications of its central zone. AbstractMesh 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.