We present a method for constructing smooth n -direction fields (line fields, cross fields, etc .) on surfaces that is an order of magnitude faster than state-of-the-art methods, while still producing fields of equal or better quality. Fields produced by the method are globally optimal in the sense that they minimize a simple, well-defined quadratic smoothness energy over all possible configurations of singularities (number, location, and index). The method is fully automatic and can optionally produce fields aligned with a given guidance field such as principal curvature directions. Computationally the smoothest field is found via a sparse eigenvalue problem involving a matrix similar to the cotan-Laplacian. When a guidance field is present, finding the optimal field amounts to solving a single linear system.
Stripe patterns are ubiquitous in nature, describing macroscopic phenomena such as stripes on plants and animals, down to material impurities on the atomic scale. We propose a method for synthesizing stripe patterns on triangulated surfaces, where singularities are automatically inserted in order to achieve userspecified orientation and line spacing. Patterns are characterized as global minimizers of a convex-quadratic energy which is welldefined in the smooth setting. Computation amounts to finding the principal eigenvector of a symmetric positive-definite matrix with the same sparsity as the standard graph Laplacian. The resulting patterns are globally continuous, and can be applied to a variety of tasks in design and texture synthesis.
We describe a new approach for the purely Eulerian simulation of incompressible fluids. In it, the fluid state is represented by a C 2 -valued wave function evolving under the Schrödinger equation subject to incompressibility constraints. The underlying dynamical system is Hamiltonian and governed by the kinetic energy of the fluid together with an energy of Landau-Lifshitz type. The latter ensures that dynamics due to thin vortical structures, all important for visual simulation, are faithfully reproduced. This enables robust simulation of intricate phenomena such as vortical wakes and interacting vortex filaments, even on modestly sized grids. Our implementation uses a simple splitting method for time integration, employing the FFT for Schrödinger evolution as well as constraint projection. Using a standard penalty method we also allow arbitrary obstacles. The resulting algorithm is simple, unconditionally stable, and efficient. In particular it does not require any Lagrangian techniques for advection or to counteract the loss of vorticity. We demonstrate its use in a variety of scenarios, compare it with experiments, and evaluate it against benchmark tests. A full implementation is included in the ancillary materials.
We study the isometric immersion problem for orientable surface triangle meshes endowed with only a metric: given the combinatorics of the mesh together with edge lengths, approximate an isometric immersion into R 3 . To address this challenge we develop a discrete theory for surface immersions into R 3 . It precisely characterizes a discrete immersion, up to subdivision and small perturbations. In particular our discrete theory correctly represents the topology of the space of immersions, i.e. , the regular homotopy classes which represent its connected components. Our approach relies on unit quaternions to represent triangle orientations and to encode, in their parallel transport, the topology of the immersion. In unison with this theory we develop a computational apparatus based on a variational principle. Minimizing a non-linear Dirichlet energy optimally finds extrinsic geometry for the given intrinsic geometry and ensures low metric approximation error. We demonstrate our algorithm with a number of applications from mathematical visualization and art directed isometric shape deformation, which mimics the behavior of thin materials with high membrane stiffness.
Discrete vector bundles are important in Physics and recently found remarkable applications in Computer Graphics. This article approaches discrete bundles from the viewpoint of Discrete Differential Geometry, including a complete classification of discrete vector bundles over finite simplicial complexes. In particular, we obtain a discrete analogue of a theorem of André Weil on the classification of hermitian line bundles. Moreover, we associate to each discrete hermitian line bundle with curvature a unique piecewise-smooth hermitian line bundle of piecewiseconstant curvature. This is then used to define a discrete Dirichlet energy which generalizes the well-known cotangent Laplace operator to discrete hermitian line bundles over Euclidean simplicial manifolds of arbitrary dimension.
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