Komplexe Linienbündel über Simplizialkomplexen

This cumulative dissertation treats practical and theoretical aspects of discrete vector bundles over simplicial complexes. Here the main focus is on discrete hermitian line bundles which, in recent years, found a number of remarkable applications in computer graphics. Two of these applications are part of this thesis: The computation of optimal n-direction fields on surfaces (Chapter 1) and the computation of stripe patterns on surfaces (Chapter 2). The developed algorithms yield, in comparison with state-of-the-art methods, results of same quality but are an order of magnitude faster. In the last chapter, motivated by their applications, discrete vector bundles are looked at from the viewpoint of discrete differential geometry (Chapter 3). This includes a complete classification of the discrete vector bundles with connection and the classification of discrete hermitian line bundles with curvature. Moreover, to each discrete hermitian line bundle with curvature is assigned a unique piecewise-smooth hermitian line bundle with connection. This leads to a generalization of the well-known cotangent-Laplace operator to arbitrary discrete hermitian line bundles with curvature over finite Euclidean simplicial complexes.Diese kumulative Dissertation behandelt praktische und theoretische Aspekte diskreter Vektorbündel über Simplizialkomplexen. Dabei stehen hier vor allem hermitesche Linienbündel im Vordergrund, welche in den letzten Jahren eine Reihe bemerkenswerter Anwendungen in der Computergrafik fanden. Zwei dieser Anwendungen sind dabei Teil dieser Dissertation: die Berechnung von optimalen n-Richtungsfeldern auf Flächen (Kapitel 1) und die Berechnung von Streifenmustern auf Flächen (Kapitel 2). Die entwickelten Algorithmen liefern im Vergleich mit anderen modernen Methoden Resultate von gleicher Qualität, sind aber um eine Größenordnung schneller. Motiviert durch ihre Anwendungen werden dann im letzten Kapitel diskrete Vektorbündel aus dem Blickwinkel der diskreten Differentialgeometrie betrachtet (Kapitel 3). Dies beinhaltet eine vollständige Klassifikation der diskreten Vektorbündel mit Zusammenhang und die Klassifikation der hermiteschen Linienbündel mit Krümmung. Weiter wird jedem diskreten hermiteschen Linienbündel mit Krümmung ein eindeutiges stückweise glattes Bündel mit Zusammenhang zugeordnet. Dies führt zu einer Verallgemeinerung des bekannten Kotangens-Laplace-Operators auf beliebige diskrete hermitesche Linienbündel mit Krümmung über endlichen Euklidischen Simplizialkomplexen.DFG, TRR 109, Discretization in Geometry and Dynamic

Shape from metric

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

Globally Optimal Direction Fields

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

Inside Fluids: Clebsch Maps for Visualization and Processing

Clebsch maps encode velocity fields through functions. These functions contain valuable information about the velocity field. For example, closed integral curves of the associated vorticity field are level lines of the vorticity Clebsch map. This makes Clebsch maps useful for visualization and fluid dynamics analysis. Additionally they can be used in the context of simulations to enhance flows through the introduction of subgrid vorticity. In this paper we study spherical Clebsch maps, which are particularly attractive. Elucidating their geometric structure, we show that such maps can be found as minimizers of a non-linear Dirichlet energy. To illustrate our approach we use a number of benchmark problems and apply it to numerically given flow fields. Code and a video can be found in the ACM Digital Library

Stripe patterns on surfaces

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 user-specified orientation and line spacing. Patterns are characterized as global minimizers of a convex-quadratic energy which is well-defined 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

Schrödinger's smoke

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

On bubble rings and ink chandeliers

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 the evolution of such filaments as an instance of Newtonian dynamics on a Riemannian configuration manifold we are able to extend classical work in the dynamics of vortex filaments through inclusion of viscous drag forces. The latter must be accounted for in low Reynolds number flows where they lead to significant variations in filament thickness and form an essential part of the observed dynamics. We develop and document both the underlying theory and associated practical numerical algorithms