1,721,068 research outputs found
Geometry processing from an elastic perspective
No full textTriggered by the development of new hardware, such as laser range scanners for high resolution acquisition of complex geometric objects, new graphics processors for realtime rendering and animation of extremely detailed geometric structures, and novel rapid prototyping equipment, such as 3D printers, the processing of highly resolved complex geometries has established itself as an important area of both fundamental research and impressive applications. Concepts from image processing have been picked up and carried over to curved surfaces, physically based modeling plays a central role, and aspects of computer aided geometry design have been incorporated. This paper aims at highlighting some of these developments, with a particular focus on methods related to the mechanics of thin elastic surfaces. We provide an overview of different geometric representations ranging from polyhedral surfaces over level sets to subdivision surfaces. Furthermore, with an eye on differential-geometric concepts underlying continuum mechanics, we discuss fundamental computational tasks, such as surface flows and fairing, surface deformation and matching, physical simulations, as well as spectral and modal methods in geometry processing. Finally, beyond focusing on single shapes, we describe how spaces of shapes can be investigated using concepts from Riemannian geometry
Variational Convergence of Discrete Minimal Surfaces
No full textBuilding on and extending tools from variational analysis, we prove Kuratowski convergence of sets of simplicial area minimizers to minimizers of the smooth Douglas-Plateau problem under simplicial refinement. This convergence is with respect to a topology that is stronger than uniform convergence of both positions and surface normals
SIGGRAPH ASIA 2008 COURSE NOTES on Discrete Differential Geometry
No full textThis volume documents the full day course Discrete Differential Geometry: An Applied Introduction at SIGGRAPH Asia 2008 in Singapore on 12 December 2008. These notes supplement the lectures given by Mathieu Desbrun, Peter Schröder, and Max Wardetzky. These notes include contributions by Miklos Bergou, Mathieu Desbrun, Sharif Elcott, Akash Garg, Eitan Grinspun, David Harmon, Eva Kanso, Felix Kälberer, Saurabh Mathur, Ulrich Pinkall, Peter Schröder, Adrian Secord, Boris Springborn, Ari Stern, John M. Sullivan, Yiying Tong, Max Wardetzky, and Denis Zorin, and build on the ideas of many others
Blind Dates für die Wissenschaft
No full textWissenschaftlicher Austausch ist eine wichtige Basis für erfolgreiche und innovative Forschung. Das klassische Format von Tagungen und Workshops im Wissenschaftsbetrieb stößt hier immer wieder an seine Grenzen. Insbesondere ist es für den wissenschaftlichen Nachwuchs häufig schwierig, von Vorträgen aus anderen Fachgebieten zu profitieren und so in einen sinnvollen Austausch mit Kolleginnen und Kollegen treten zu können. Für das Berliner DFG-Forschungszentrum MATHEON entwickelten wir ein neues Konzept, um einen vertieften wissenschaftlichen Austausch besser zu fördern
On the convergence of metric and geometric properties of polyhedral surfaces
No full textWe provide conditions for convergence of polyhedral surfaces and their discrete geometric properties to smooth surfaces embedded in Euclidean 3-space. Under the assumption of convergence of surfaces in Hausdorff distance, we show that convergence of the following properties are equivalent: surface normals, surface area, metric tensors, and Laplace–Beltrami operators.Additionally, we derive convergence of minimizing geodesics, mean curvature vectors, and solutions to the Dirichlet problem
Algebraic Topology on Polyhedral Surfaces from Finite Elements
No full textWe report on a development using piecewise constant vector fields (or one-forms) on compact polyhedral surfaces. The function spaces corresponding to a discrete Hodge decomposition then turn out to be a mixture of conforming and nonconforming linear finite elements. For sequences of polyhedral surfaces whose positions and normals converge to the positions and normals of an embedded compact smooth surface, we report on a convergence result for the corresponding discrete Hodge decompositions and Hodge star operators
Heat kernel asymptotics for scaling limits of isoradial graphs
No full textWe consider the asymptotics of the discrete heat kernel on isoradial graphs for the case where the time and the edge lengths tend to zero simultaneously. Depending on the asymptotic ratio between time and edge lengths, we show that two different regimes arise, corresponding to the short-time asymptotics of the heat kernel on (i) Euclidean spaces and (ii) on graphs
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