article published yes Christian Hafner author 400429CC-F248-11E8-B48F-1D18A9856A87 Mickaël Ly author 6340d7f0-b48d-11eb-b10d-b7487e71d9f1 Christopher J Wojtan author 3C61F1D2-F248-11E8-B48F-1D18A9856A870000-0001-6646-5546 ChWo department Computational Discovery of Numerical Algorithms for Animation and Simulation of Natural Phenomena project 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. https://research-explorer.ista.ac.at/download/17203/17204/sif-final.pdf application/pdfno https://research-explorer.ista.ac.at/download/17203/17205/sif-supp-final.pdf application/pdfno https://research-explorer.ista.ac.at/download/17203/17276/sif-video-final.mp4 video/mp4no Association for Computing Machinery2024Denver, Colorado eng Topology OptimizationMass MomentsComputational Geometry 0730-0301 1557-7368 00128927090004510.1145/3658194 434 Hafner, C., Ly, M., &#38; Wojtan, C. (2024). Spin-it faster: Quadrics solve all topology optimization problems that depend only on mass moments. <i>Transactions on Graphics</i>. Denver, Colorado: Association for Computing Machinery. <a href="https://doi.org/10.1145/3658194">https://doi.org/10.1145/3658194</a> C. Hafner, M. Ly, and C. Wojtan, “Spin-it faster: Quadrics solve all topology optimization problems that depend only on mass moments,” <i>Transactions on Graphics</i>, vol. 43, no. 4. Association for Computing Machinery, 2024. Hafner C, Ly M, Wojtan C. Spin-it faster: Quadrics solve all topology optimization problems that depend only on mass moments. <i>Transactions on Graphics</i>. 2024;43(4). doi:<a href="https://doi.org/10.1145/3658194">10.1145/3658194</a> Hafner, Christian, et al. “Spin-It Faster: Quadrics Solve All Topology Optimization Problems That Depend Only on Mass Moments.” <i>Transactions on Graphics</i>, vol. 43, no. 4, 78, Association for Computing Machinery, 2024, doi:<a href="https://doi.org/10.1145/3658194">10.1145/3658194</a>. Hafner, Christian, Mickaël Ly, and Chris Wojtan. “Spin-It Faster: Quadrics Solve All Topology Optimization Problems That Depend Only on Mass Moments.” <i>Transactions on Graphics</i>. Association for Computing Machinery, 2024. <a href="https://doi.org/10.1145/3658194">https://doi.org/10.1145/3658194</a>. C. Hafner, M. Ly, C. Wojtan, Transactions on Graphics 43 (2024). Hafner C, Ly M, Wojtan C. 2024. Spin-it faster: Quadrics solve all topology optimization problems that depend only on mass moments. Transactions on Graphics. 43(4), 78. 172032024-07-05T12:08:57Z2026-10-01T09:30:01Z