We describe a new approach for the purely Eulerian simulation of incompressible fluids. In it, fluid state is represented by C 2 -valued wave function evolving under Schrödinger equation subject to incompressibility constraints. The underlying dynamical system Hamiltonian and governed kinetic energy together with an Landau-Lifshitz type. latter ensures that dynamics due thin vortical structures, all important visual simulation, are faithfully reproduced. This enables robust intricate...

We propose a method for computing global Chebyshev nets on triangular meshes. formulate the corresponding parameterization problem in terms of commuting PolyVector fields, and design an efficient optimization to solve it. compute, first time, with automatically-placed singularities, demonstrate realizability our approach using real material.

The dynamics of evolving fluid films in the viscous Stokes limit is relevant to various applications, such as modelling lipid bilayers cells. While governing equations were formulated by Scriven (1960), solving for flow a deformable surface with arbitrary shape and topology has remained challenge. In this study, we present straightforward discrete model based on variational principles address long-standing problem. We replace classical equations, which are expressed tensor calculus local...

Recent advances in cell biology and experimental techniques using reconstituted extracts have generated significant interest understanding how geometry topology influence active fluid dynamics. In this work, we present a comprehensive continuum theory computational method to explore the dynamics of nematic fluids on arbitrary surfaces without topological constraints. The velocity order parameter are represented as sections complex line bundle two-manifold. We introduce Levi–Civita connection...

The animation of delicate vortical structures gas and liquids has been great interest in computer graphics. However, common velocity-based fluid solvers can damp the flow, while vorticity-based suffer from performance drawbacks. We propose a new solver derived reformulated Euler equation using covectors. Our method generates rich vortex dynamics by an advection process that respects Kelvin circulation theorem. numerical algorithm requires only small local adjustment to existing...

We study the isometric immersion problem for orientable surface triangle meshes endowed with only a metric: given combinatorics of mesh together edge lengths, approximate an into R 3 . To address this challenge we develop discrete theory immersions It precisely characterizes immersion, up to subdivision and small perturbations. In particular our correctly represents topology space immersions, i.e. , regular homotopy classes which represent its connected components. Our approach relies on...

Recent techniques have been successful in reconstructing surfaces as level sets of learned functions (such signed distance fields) parameterized by deep neural networks. Many these methods, however, learn only closed and are unable to reconstruct shapes with boundary curves. We propose a hybrid shape representation that combines explicit curves implicit interiors. Using machinery from geometric measure theory, we parameterize currents using networks use stochastic gradient descent solve...

The vorticity-streamfunction formulation for incompressible inviscid fluids is the basis many fluid simulation methods in computer graphics, including vortex methods, streamfunction solvers, spectral and Monte Carlo methods. We point out that current setups are insufficient at simulating on general non-simply-connected domains. This issue critical practice, as obstacles, periodic boundaries, nonzero genus can all make domain multiply connected. These scenarios introduce nontrivial cohomology...

Clebsch maps encode velocity fields through functions. These functions contain valuable information about the field. For example, closed integral curves of associated vorticity field are level lines map. This makes useful for visualization and fluid dynamics analysis. Additionally they can be used in context simulations to enhance flows introduction subgrid vorticity. In this paper we study spherical maps, which particularly attractive. Elucidating their geometric structure, show that such...

We introduce variable thickness, viscous vortex filaments. These can model such varied phenomena as underwater bubble rings or the intricate "chandeliers" formed by ink dropping into fluid. Treating evolution of filaments an instance Newtonian dynamics on a Riemannian configuration manifold we are able to extend classical work in through inclusion drag forces. The latter must be accounted for low Reynolds number flows where they lead significant variations filament thickness and form...

Solving partial differential equations (PDEs) on infinite domains has been a challenging task in physical simulations and geometry processing. We introduce general technique to transform PDE problem an unbounded domain bounded domain. Our method uses the Kelvin Transform, which essentially inverts distance from origin. However, naive application of this coordinate mapping can still result singularity at origin transformed show that by factoring desired solution into product analytically...

Smooth curves and surfaces can be characterized as minimizers of squared curvature bending energies subject to constraints. In the univariate case with an isometry (length) constraint this leads classic non-linear splines. For surfaces, is too rigid a instead one asks for Willmore (squared mean curvature) energy conformality constraint. We present efficient algorithm (conformally) constrained using triangle meshes arbitrary topology or without boundary. Our conformal class based on discrete...

Conformal deformations are infinitesimal scale-rotations, which can be parameterized by quaternions. The condition that such a quaternion field gives rise to conformal deformation is nonlinear and in any case only admits Möbius transformations as solutions. We propose particular decoupling of scaling rotation allows us find near minimizers quadratic, convex Dirichlet energy. Applied tetrahedral meshes we with low quasiconformal distortion the principal eigenvector (quaternionic) Laplace...

We describe a new algorithm that solves classical geometric problem: Find surface of minimal area bordered by an arbitrarily prescribed boundary curve. Existing numerical methods face challenges due to the non-convexity problem. Using representation curves and surfaces via differential forms on ambient space, we reformulate this problem as convex optimization. This change variables overcomes many difficulties in previous attempts allows us find global minimum across all possible topologies....

In this paper, we consider using barrier function-based approaches for the safe control problem in stochastic systems. presence of uncertainties, a myopic controller that ensures probability infinitesimal time intervals may suffer from accumulation unsafe over and result small long-term probability. Meanwhile, increasing outlook horizon lead to significant computation burdens delayed reactions, which also compromises safety. To tackle challenge, define new notion forward invariance on...

The demand for a more advanced multivariable calculus has rapidly increased in computer graphics research, such as physical simulation, geometry synthesis, and differentiable rendering. Researchers often have to turn references outside of research study identities the Reynolds Transport Theorem or geometric relationship between stress strain tensors. This course presents comprehensive introduction exterior calculus, which covers many these topics geometrically intuitive manner. targets...

Accurate quantification of safety is essential for the design autonomous systems. In this paper, we present a methodology to characterize exact probabilities associated with invariance and recovery in safe control. We consider stochastic control system where barrier functions, gradient-based methods, certificates are used constrain actions validate safety. derive probability distributions minimum maximum function values during any time interval first entry exit times from super level sets...

This paper presents a new representation of curve dynamics, with applications to vortex filaments in fluid dynamics. Instead representing these explicit geometry and Lagrangian equations motion, we represent curves implicitly co-dimensional 2 level set description. Our implicit admits several redundant mathematical degrees freedom both the configuration dynamics curves, which can be tailored specifically improve numerical robustness, contrast naive approaches for that suffer from...

Video-based glint-free eye tracking commonly estimates gaze direction based on the pupil center. The boundary of is fitted with an ellipse and euclidean center in image taken as pupil. However, generally not mapped to by projective camera transformation. This error resulting from using a point that true directly affects accuracy. We investigate underlying geometric problem determining circular object its image. main idea exploit two concentric circles -- application scenario these are iris....

Simulation of stellar atmospheres, such as that our own sun, is a common task in CGI for scientific visualization, movies and games. A fibrous volumetric texture visually dominant feature the solar corona---the plasma extends from surface into space. These coronal fibers can be modeled magnetic filaments whose shape governed by magnetohydrostatic equation. The provide Lagrangian curve representation their initial configuration prescribed an artist or generated flux given scalar on sun's...

We describe a new algorithm that solves classical geometric problem: Find surface of minimal area bordered by an arbitrarily prescribed boundary curve. Existing numerical methods face challenges due to the non-convexity problem. Using representation curves and surfaces via differential forms on ambient space, we reformulate this problem as convex optimization. This change variables overcomes many difficulties in previous attempts allows us find global minimum across all possible topologies....

This paper addresses the design of safety certificates for stochastic systems, with a focus on ensuring long-term through fast real-time control. In environments, set invariance-based methods that restrict probability risk events in infinitesimal time intervals may exhibit significant risks due to cumulative uncertainties/risks. On other hand, reachability-based approaches account future require prohibitive computation decision making. To overcome this challenge involving stringent vs....

Directional fields, including unit vector, line, and cross are essential tools in the geometry processing toolkit. The topology of directional fields is characterized by their singularities. While singularities play an important role downstream applications such as meshing, existing methods for computing either require them to be specified advance, ignore altogether, or treat zeros a relaxed field. ill-defined at singularities, graphs with well-defined surfaces circle bundle. By lifting...

We propose Coadjoint Orbit FLIP (CO-FLIP), a high order accurate, structure preserving fluid simulation method in the hybrid Eulerian-Lagrangian framework. start with Hamiltonian formulation of incompressible Euler Equations, and then, using local, explicit, divergence free interpolation, construct modified system that governs our discrete flow. The resulting discretization, when paired geometric time integration scheme, is energy circulation (formally flow evolves on coadjoint orbit)...