@inproceedings{22129, abstract = {Inside–outside classification is widely used for geometry processing tasks such as surface reconstruction, geometry completion, and calculating signed distance fields. We introduce a new integral formulation of this problem, which assigns confidence scores that points are inside or outside, given incomplete boundary geometry. Even though our geometric construction does not appear in previous work, we show that it is unexpectedly linked to both the well-established generalized winding number (GWN) and pseudonormal methods for geometry completion, and it provably reduces to either one of them for specific values of a control parameter. The results obtained with our method frequently outperform screened Poisson surface reconstruction (PSR), GWN, and the pseudonormal method in terms of quality, and are at least on par with them on all of our examples. Unlike these methods, our algorithm naturally extends to the multi-label setting, in which regions with an arbitrary number of colors or physical materials can be reconstructed, and non-manifold features such as T-junctions may appear in the interface and boundary geometry}, author = {Wei, Ziyu and Hafner, Christian and Kalinov, Aleksei and Synak, Peter and Wojtan, Christopher J}, booktitle = {Computer Graphics Forum}, location = {Bern, Switzerland}, number = {5}, publisher = {Wiley}, title = {{Circles of confidence for multi-label geometry completion}}, doi = {10.1111/cgf.70516}, volume = {45}, year = {2026}, } @article{21923, abstract = {The appearance of simulated natural phenomena heavily depends on the way surfaces are textured. However, applying texture maps to dynamic deformable surfaces presents a significant challenge, due to ever-shifting differences in length scales involved. When these surfaces move and advect the texture along with them, their final appearance degrades as deformed regions dramatically distort their texture map. Modifications to the texture directly at the pixel level in response to the deformation may introduce ghosting artifacts and look unnatural. In the real world, the appearance of surface details on a deforming material changes through the interplay of physical processes such as rupturing, exposure of internal structure, or wrinkling. Motivated by these behaviors, in this work we explore how physical principles can guide the texturing methods based on the measure of surface deformation. We present two novel wave-based procedural texturing algorithms which reproduce common physical properties like advection and self-similarity, enabling the plausible animation of deforming objects with extreme texture map distortions. Our algorithms are fully procedural, require no actual physics simulation, and store no state or history of deformation besides the input UV map, making them highly parallelizable on the GPU and efficient enough for real-time applications. We show the versatility of the method by animating physical phenomena with extreme deformations such as flowing lava, stretching putty and outpouring sludge.}, author = {Kalinov, Aleksei and Ly, Mickaël and Hafner, Christian and Wojtan, Christopher J}, issn = {0730-0301}, journal = {ACM Transactions on Graphics}, keywords = {Procedural animation}, location = {Los Angeles, CA, United States}, number = {4}, publisher = {ACM}, title = {{Physics-inspired procedural texturing of extremely deformable surfaces}}, doi = {10.1145/3811353}, volume = {45}, year = {2026}, } @article{17203, abstract = {The behavior of a rigid body primarily depends on its mass moments, which consist of the mass, center of mass, and moments of inertia. It is possible to manipulate these quantities without altering the geometric appearance of an object by introducing cavities in its interior. Algorithms that find cavities of suitable shapes and sizes have enabled the computational design of spinning tops, yo-yos, wheels, buoys, and statically balanced objects. Previous work is based, for example, on topology optimization on voxel grids, which introduces a large number of optimization variables and box constraints, or offset surface computation, which cannot guarantee that solutions to a feasible problem will always be found. In this work, we provide a mathematical analysis of constrained topology optimization problems that depend only on mass moments. This class of problems covers, among others, all applications mentioned above. Our main result is to show that no matter the outer shape of the rigid body to be optimized or the optimization objective and constraints considered, the optimal solution always features a quadric-shaped interface between material and cavities. This proves that optimal interfaces are always ellipsoids, hyperboloids, paraboloids, or one of a few degenerate cases, such as planes. This insight lets us replace a difficult topology optimization problem with a provably equivalent non-linear equation system in a small number (<10) of variables, which represent the coefficients of the quadric. This system can be solved in a few seconds for most examples, provides insights into the geometric structure of many specific applications, and lets us describe their solution properties. Finally, our method integrates seamlessly into modern fabrication workflows because our solutions are analytical surfaces that are native to the CAD domain.}, author = {Hafner, Christian and Ly, Mickaël and Wojtan, Christopher J}, issn = {1557-7368}, journal = {Transactions on Graphics}, keywords = {Topology Optimization, Mass Moments, Computational Geometry}, location = {Denver, Colorado}, number = {4}, publisher = {Association for Computing Machinery}, title = {{Spin-it faster: Quadrics solve all topology optimization problems that depend only on mass moments}}, doi = {10.1145/3658194}, volume = {43}, year = {2024}, } @phdthesis{12897, abstract = {Inverse design problems in fabrication-aware shape optimization are typically solved on discrete representations such as polygonal meshes. This thesis argues that there are benefits to treating these problems in the same domain as human designers, namely, the parametric one. One reason is that discretizing a parametric model usually removes the capability of making further manual changes to the design, because the human intent is captured by the shape parameters. Beyond this, knowledge about a design problem can sometimes reveal a structure that is present in a smooth representation, but is fundamentally altered by discretizing. In this case, working in the parametric domain may even simplify the optimization task. We present two lines of research that explore both of these aspects of fabrication-aware shape optimization on parametric representations. The first project studies the design of plane elastic curves and Kirchhoff rods, which are common mathematical models for describing the deformation of thin elastic rods such as beams, ribbons, cables, and hair. Our main contribution is a characterization of all curved shapes that can be attained by bending and twisting elastic rods having a stiffness that is allowed to vary across the length. Elements like these can be manufactured using digital fabrication devices such as 3d printers and digital cutters, and have applications in free-form architecture and soft robotics. We show that the family of curved shapes that can be produced this way admits geometric description that is concise and computationally convenient. In the case of plane curves, the geometric description is intuitive enough to allow a designer to determine whether a curved shape is physically achievable by visual inspection alone. We also present shape optimization algorithms that convert a user-defined curve in the plane or in three dimensions into the geometry of an elastic rod that will naturally deform to follow this curve when its endpoints are attached to a support structure. Implemented in an interactive software design tool, the rod geometry is generated in real time as the user edits a curve and enables fast prototyping. The second project tackles the problem of general-purpose shape optimization on CAD models using a novel variant of the extended finite element method (XFEM). Our goal is the decoupling between the simulation mesh and the CAD model, so no geometry-dependent meshing or remeshing needs to be performed when the CAD parameters change during optimization. This is achieved by discretizing the embedding space of the CAD model, and using a new high-accuracy numerical integration method to enable XFEM on free-form elements bounded by the parametric surface patches of the model. Our simulation is differentiable from the CAD parameters to the simulation output, which enables us to use off-the-shelf gradient-based optimization procedures. The result is a method that fits seamlessly into the CAD workflow because it works on the same representation as the designer, enabling the alternation of manual editing and fabrication-aware optimization at will.}, author = {Hafner, Christian}, isbn = {978-3-99078-031-2}, issn = {2663-337X}, pages = {180}, publisher = {Institute of Science and Technology Austria}, title = {{Inverse shape design with parametric representations: Kirchhoff Rods and parametric surface models}}, doi = {10.15479/at:ista:12897}, year = {2023}, } @article{12972, abstract = {Embroidery is a long-standing and high-quality approach to making logos and images on textiles. Nowadays, it can also be performed via automated machines that weave threads with high spatial accuracy. A characteristic feature of the appearance of the threads is a high degree of anisotropy. The anisotropic behavior is caused by depositing thin but long strings of thread. As a result, the stitched patterns convey both color and direction. Artists leverage this anisotropic behavior to enhance pure color images with textures, illusions of motion, or depth cues. However, designing colorful embroidery patterns with prescribed directionality is a challenging task, one usually requiring an expert designer. In this work, we propose an interactive algorithm that generates machine-fabricable embroidery patterns from multi-chromatic images equipped with user-specified directionality fields.We cast the problem of finding a stitching pattern into vector theory. To find a suitable stitching pattern, we extract sources and sinks from the divergence field of the vector field extracted from the input and use them to trace streamlines. We further optimize the streamlines to guarantee a smooth and connected stitching pattern. The generated patterns approximate the color distribution constrained by the directionality field. To allow for further artistic control, the trade-off between color match and directionality match can be interactively explored via an intuitive slider. We showcase our approach by fabricating several embroidery paths.}, author = {Liu, Zhenyuan and Piovarci, Michael and Hafner, Christian and Charrondiere, Raphael and Bickel, Bernd}, issn = {1467-8659}, journal = {Computer Graphics Forum}, keywords = {embroidery, design, directionality, density, image}, location = {Saarbrucken, Germany}, number = {2}, pages = {397--409}, publisher = {Wiley}, title = {{Directionality-aware design of embroidery patterns}}, doi = {10.1111/cgf.14770 }, volume = {42}, year = {2023}, } @article{13188, abstract = {The Kirchhoff rod model describes the bending and twisting of slender elastic rods in three dimensions, and has been widely studied to enable the prediction of how a rod will deform, given its geometry and boundary conditions. In this work, we study a number of inverse problems with the goal of computing the geometry of a straight rod that will automatically deform to match a curved target shape after attaching its endpoints to a support structure. Our solution lets us finely control the static equilibrium state of a rod by varying the cross-sectional profiles along its length. We also show that the set of physically realizable equilibrium states admits a concise geometric description in terms of linear line complexes, which leads to very efficient computational design algorithms. Implemented in an interactive software tool, they allow us to convert three-dimensional hand-drawn spline curves to elastic rods, and give feedback about the feasibility and practicality of a design in real time. We demonstrate the efficacy of our method by designing and manufacturing several physical prototypes with applications to interior design and soft robotics.}, author = {Hafner, Christian and Bickel, Bernd}, issn = {1557-7368}, journal = {ACM Transactions on Graphics}, keywords = {Computer Graphics, Computational Design, Computational Geometry, Shape Modeling}, number = {5}, publisher = {Association for Computing Machinery}, title = {{The design space of Kirchhoff rods}}, doi = {10.1145/3606033}, volume = {42}, year = {2023}, } @article{9817, abstract = {Elastic bending of initially flat slender elements allows the realization and economic fabrication of intriguing curved shapes. In this work, we derive an intuitive but rigorous geometric characterization of the design space of plane elastic rods with variable stiffness. It enables designers to determine which shapes are physically viable with active bending by visual inspection alone. Building on these insights, we propose a method for efficiently designing the geometry of a flat elastic rod that realizes a target equilibrium curve, which only requires solving a linear program. We implement this method in an interactive computational design tool that gives feedback about the feasibility of a design, and computes the geometry of the structural elements necessary to realize it within an instant. The tool also offers an iterative optimization routine that improves the fabricability of a model while modifying it as little as possible. In addition, we use our geometric characterization to derive an algorithm for analyzing and recovering the stability of elastic curves that would otherwise snap out of their unstable equilibrium shapes by buckling. We show the efficacy of our approach by designing and manufacturing several physical models that are assembled from flat elements.}, author = {Hafner, Christian and Bickel, Bernd}, issn = {1557-7368}, journal = {ACM Transactions on Graphics}, keywords = {Computing methodologies, shape modeling, modeling and simulation, theory of computation, computational geometry, mathematics of computing, mathematical optimization}, location = {Virtual}, number = {4}, publisher = {Association for Computing Machinery}, title = {{The design space of plane elastic curves}}, doi = {10.1145/3450626.3459800}, volume = {40}, year = {2021}, } @article{8766, abstract = {The “procedural” approach to animating ocean waves is the dominant algorithm for animating larger bodies of water in interactive applications as well as in off-line productions — it provides high visual quality with a low computational demand. In this paper, we widen the applicability of procedural water wave animation with an extension that guarantees the satisfaction of boundary conditions imposed by terrain while still approximating physical wave behavior. In combination with a particle system that models wave breaking, foam, and spray, this allows us to naturally model waves interacting with beaches and rocks. Our system is able to animate waves at large scales at interactive frame rates on a commodity PC.}, author = {Jeschke, Stefan and Hafner, Christian and Chentanez, Nuttapong and Macklin, Miles and Müller-Fischer, Matthias and Wojtan, Christopher J}, journal = {Computer Graphics forum}, location = {Online Symposium}, number = {8}, pages = {47--54}, publisher = {Wiley}, title = {{Making procedural water waves boundary-aware}}, doi = {10.1111/cgf.14100}, volume = {39}, year = {2020}, } @article{6442, abstract = {This paper investigates the use of fundamental solutions for animating detailed linear water surface waves. We first propose an analytical solution for efficiently animating circular ripples in closed form. We then show how to adapt the method of fundamental solutions (MFS) to create ambient waves interacting with complex obstacles. Subsequently, we present a novel wavelet-based discretization which outperforms the state of the art MFS approach for simulating time-varying water surface waves with moving obstacles. Our results feature high-resolution spatial details, interactions with complex boundaries, and large open ocean domains. Our method compares favorably with previous work as well as known analytical solutions. We also present comparisons between our method and real world examples.}, author = {Schreck, Camille and Hafner, Christian and Wojtan, Christopher J}, journal = {ACM Transactions on Graphics}, number = {4}, publisher = {ACM}, title = {{Fundamental solutions for water wave animation}}, doi = {10.1145/3306346.3323002}, volume = {38}, year = {2019}, } @article{7117, abstract = {We propose a novel generic shape optimization method for CAD models based on the eXtended Finite Element Method (XFEM). Our method works directly on the intersection between the model and a regular simulation grid, without the need to mesh or remesh, thus removing a bottleneck of classical shape optimization strategies. This is made possible by a novel hierarchical integration scheme that accurately integrates finite element quantities with sub-element precision. For optimization, we efficiently compute analytical shape derivatives of the entire framework, from model intersection to integration rule generation and XFEM simulation. Moreover, we describe a differentiable projection of shape parameters onto a constraint manifold spanned by user-specified shape preservation, consistency, and manufacturability constraints. We demonstrate the utility of our approach by optimizing mass distribution, strength-to-weight ratio, and inverse elastic shape design objectives directly on parameterized 3D CAD models.}, author = {Hafner, Christian and Schumacher, Christian and Knoop, Espen and Auzinger, Thomas and Bickel, Bernd and Bächer, Moritz}, issn = {0730-0301}, journal = {ACM Transactions on Graphics}, number = {6}, publisher = {ACM}, title = {{X-CAD: Optimizing CAD Models with Extended Finite Elements}}, doi = {10.1145/3355089.3356576}, volume = {38}, year = {2019}, }