TY - GEN AB - We present a formula for the signed area of a spherical polygon via prequantization. In contrast to the traditional formula based on the Gauss-Bonnet theorem that requires measuring angles, the new formula mimics Green's theorem and is applicable to a wider range of degenerate spherical curves and polygons. AU - Chern, Albert AU - Ishida, Sadashige ID - 12846 T2 - arXiv TI - Area formula for spherical polygons via prequantization ER - TY - JOUR AB - We introduce a compact, intuitive procedural graph representation for cellular metamaterials, which are small-scale, tileable structures that can be architected to exhibit many useful material properties. Because the structures’ “architectures” vary widely—with elements such as beams, thin shells, and solid bulks—it is difficult to explore them using existing representations. Generic approaches like voxel grids are versatile, but it is cumbersome to represent and edit individual structures; architecture-specific approaches address these issues, but are incompatible with one another. By contrast, our procedural graph succinctly represents the construction process for any structure using a simple skeleton annotated with spatially varying thickness. To express the highly constrained triply periodic minimal surfaces (TPMS) in this manner, we present the first fully automated version of the conjugate surface construction method, which allows novices to create complex TPMS from intuitive input. We demonstrate our representation’s expressiveness, accuracy, and compactness by constructing a wide range of established structures and hundreds of novel structures with diverse architectures and material properties. We also conduct a user study to verify our representation’s ease-of-use and ability to expand engineers’ capacity for exploration. AU - Makatura, Liane AU - Wang, Bohan AU - Chen, Yi-Lu AU - Deng, Bolei AU - Wojtan, Christopher J AU - Bickel, Bernd AU - Matusik, Wojciech ID - 14628 IS - 5 JF - ACM Transactions on Graphics KW - Computer Graphics and Computer-Aided Design SN - 0730-0301 TI - Procedural metamaterials: A unified procedural graph for metamaterial design VL - 42 ER - TY - GEN AB - We present a discretization of the dynamic optimal transport problem for which we can obtain the convergence rate for the value of the transport cost to its continuous value when the temporal and spatial stepsize vanish. This convergence result does not require any regularity assumption on the measures, though experiments suggest that the rate is not sharp. Via an analysis of the duality gap we also obtain the convergence rates for the gradient of the optimal potentials and the velocity field under mild regularity assumptions. To obtain such rates we discretize the dual formulation of the dynamic optimal transport problem and use the mature literature related to the error due to discretizing the Hamilton-Jacobi equation. AU - Ishida, Sadashige AU - Lavenant, Hugo ID - 14703 KW - Optimal transport KW - Hamilton-Jacobi equation KW - convex optimization T2 - arXiv TI - Quantitative convergence of a discretization of dynamic optimal transport using the dual formulation ER - TY - JOUR AB - This paper introduces a novel method for simulating large bodies of water as a height field. At the start of each time step, we partition the waves into a bulk flow (which approximately satisfies the assumptions of the shallow water equations) and surface waves (which approximately satisfy the assumptions of Airy wave theory). We then solve the two wave regimes separately using appropriate state-of-the-art techniques, and re-combine the resulting wave velocities at the end of each step. This strategy leads to the first heightfield wave model capable of simulating complex interactions between both deep and shallow water effects, like the waves from a boat wake sloshing up onto a beach, or a dam break producing wave interference patterns and eddies. We also analyze the numerical dispersion created by our method and derive an exact correction factor for waves at a constant water depth, giving us a numerically perfect re-creation of theoretical water wave dispersion patterns. AU - Jeschke, Stefan AU - Wojtan, Christopher J ID - 14240 IS - 4 JF - ACM Transactions on Graphics SN - 0730-0301 TI - Generalizing shallow water simulations with dispersive surface waves VL - 42 ER - TY - GEN AU - Chen, Yi-Lu AU - Ly, Mickaël AU - Wojtan, Christopher J ID - 14748 SN - 9798400702686 T2 - Proceedings of the ACM SIGGRAPH/Eurographics Symposium on Computer Animation TI - Unified treatment of contact, friction and shock-propagation in rigid body animation ER -