Max Wardetzky, Assistant Professor

Since August 2008 I am in Göttingen, where I am heading the Discrete Differential Geometry Lab, established as part of the university's excellence initiative.

Research interests:

Differential Geometry (smooth and discrete), Numerical Analysis, Physical Simulation, Computer Graphics.

Contact:

Institute of Num. and Appl. Math University of Göttingen Lotzestr. 16-18 37083 Göttingen, Germany

Vox: +49 (0)551 39-22235 Fax: +49 (0)551 39-3944 Net: lastname (at) math.uni-goettingen.de

Preprints:

Uniform Convergence of Discrete Curvatures from Nets of Curvature Lines Ulrich Bauer, Konrad Polthier, Max Wardetzky, submitted to Discrete and Compuational Geometry. Abstract: We study “Steiner-type” discrete curvatures computed from nets of curvature lines on a given smooth surface, and prove their uniform pointwise convergence to smooth principal curvatures. We provide explicit error bounds, with constants depending only on the limit surface and the shape regularity of the discrete net. [arxiv]

Publications:

	Discrete Elastic Rods Miklos Bergou, Max Wardetzky, Stephen Robinson, Basile Audoly, Eitan Grinspun, ACM Transaction on Graphics 27:3 (SIGGRAPH) 2008. Abstract: We present a discrete treatment of adapted framed curves, parallel transport, and holonomy, thus establishing the language for a discrete geometric model of thin flexible rods with arbitrary cross section and undeformed configuration. Our approach differs from existing simulation techniques in the graphics and mechanics literature both in the kinematic description - we represent the material frame by its angular deviation from the natural Bishop frame - as well as in the dynamical treatment - we treat the centerline as dynamic and the material frame as quasistatic. Additionally, we describe a manifold projection method for coupling rods to rigid-bodies and simultaneously enforcing rod inextensibility. The use of quasistatics and constraints provides an efficient treatment for stiff twisting and stretching modes; at the same time, we retain the dynamic bending of the centerline and accurately reproduce the coupling between bending and twisting modes. We validate the discrete rod model via quantitative buckling, stability, and coupled-mode experiments, and via qualitative knot-tying comparisons. [pdf] [Video]
	Convergence of the Cotangent Formula: An Overview Max Wardetzky, in "Discrete Differential Geometry" (A. I. Bobenko, John M. Sullivan, Peter Schröder, Günter Ziegler, eds.), Birkhäuser Basel, 2008. Abstract: The cotangent formula constitutes an intrinsic discretization of the Laplace- Beltrami operator on polyhedral surfaces in a finite element sense. This note gives an overview of approximation and convergence properties of discrete Laplacians and mean curvature vectors for polyhedral surfaces located in the vicinity of a smooth surface in Euclidean 3-space. In particular, we show that mean curvature vectors converge in the sense of distributions, but fail to converge in L^2. [pdf]
	Discrete Laplace operators: No free lunch Max Wardetzky, Saurabh Mathur, Felix Kälberer, Eitan Grinspun, Symposium on Geometry Processing, 2007, pp. 33-37. Abstract: Discrete Laplace operators are ubiquitous in applications spanning geometric modeling to simulation. For robustness and efficiency, many applications require discrete operators that retain key structural properties inherent to the continuous setting. Building on the smooth setting, we present a set of natural properties for discrete Laplace operators for triangular surface meshes. We prove an important theoretical limitation: discrete Laplacians cannot satisfy all natural properties; retroactively, this explains the diversity of existing discrete Laplace operators. Finally, we present a family of operators that includes and extends well-known and widely-used operators. [pdf]
	TRACKS: Toward Directable Thin Shells Miklos Bergou, Saurabh Mathur, Max Wardetzky, Eitan Grinspun, ACM Transaction on Graphics 26:3 (SIGGRAPH) 2007. Abstract: We combine the often opposing forces of artistic freedom and mathematical determinism to enrich a given animation or simulation of a surface with physically based detail. We present a process called tracking, which takes as input a rough animation or simulation and enhances it with physically simulated detail. Building on the foundation of constrained Lagrangian mechanics, we propose weak-form constraints for tracking the input motion. This method allows the artist to choose where to add details such as characteristic wrinkles and folds of various thin shell materials and dynamical effects of physical forces. We demonstrate multiple applications ranging from enhancing an artist's animated character to guiding a simulated inanimate object. [pdf] [Video]
	Cubic Shells Akah Garg, Eitan Grinspun, Max Wardetzky, Denis Zorin, Symposium on Computer Animation, 2007, pp. 91-98. Abstract: Hinge-based bending models are widely used in the physically-based animation of cloth, thin plates and shells. We propose a hinge-based model that is simpler to implement, more efficient to compute, and offers a greater number of effective material parameters than existing models. Our formulation builds on two mathematical observations: (a) the bending energy of curved flexible surfaces can be expressed as a cubic polynomial if the surface does not stretch; (b) a general class of anisotropic materials - those that are orthotropic - is captured by appropriate choice of a single stiffness per hinge. Our contribution impacts a general range of surface animation applications, from isotropic cloth and thin plates to orthotropic fracturing thin shells. [pdf] [Video]
	Discrete Quadratic Curvature Energies Max Wardetzky, Miklos Bergou, David Harmon, Denis Zorin, Eitan Grinspun, Computer Aided Geometric Design (CAGD) 24, 2007, pp. 499-518. Abstract: We present a family of discrete isometric bending models (IBMs) for triangulated surfaces in 3-space. These models are derived from an axiomatic treatment of discrete Laplace operators, using these operators to obtain linear models for discrete mean curvature from which bending energies are assembled. Under the assumption of isometric surface deformations we show that these energies are quadratic in surface positions. The corresponding linear energy gradients and constant energy Hessians constitute an efficient model for computing bending forces and their derivatives, enabling fast time-integration of cloth dynamics with a two- to three-fold net speedup over existing nonlinear methods, and near-interactive rates for Willmore smoothing of large meshes. [pdf] [Video]
	On the Convergence of Metric and Geometric Properties of Polyhedral Surfaces Klaus Hildebrandt, Konrad Polthier, Max Wardetzky, in Geometriae Dedicata 123, 2006, pp. 89-112. Abstract: We 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. [pdf]
	A Quadratic Bending Model for Inextensible Surfaces Miklos Bergou, Max Wardetzky, David Harmon, Denis Zorin, Eitan Grinspun, Symposium on Geometry Processing, 2006, pp. 227-230. Abstract: Efficient computation of curvature-based energies is important for practical implementations of geometric modeling and physical simulation applications. Building on a simple geometric observation, we provide a version of a curvature-based energy expressed in terms of the Laplace operator acting on the embedding of the surface. The corresponding energy, being quadratic in positions, gives rise to a constant Hessian in the context of isometric deformations. The resulting isometric bending model is shown to significantly speed up common cloth solvers, and when applied to geometric modeling situations built on Willmore flow to provide runtimes which are close to interactive rates. [pdf] [Video]
	Smooth Feature Lines on Surface Meshes Klaus Hildebrandt, Konrad Polthier, Max Wardetzky, Symposium on Geometry Processing, 2005, pp. 85-90. Abstract: Feature lines are salient surface characteristics. Their definition involves third and fourth order surface derivatives. This often yields to unpleasantly rough and squiggly feature lines since third order derivatives are highly sensitive against unwanted surface noise. The present work proposes two novel concepts for a more stable algorithm producing visually more pleasing feature lines: First, a new computation scheme based on discrete differential geometry is presented, avoiding costly computations of higher order approximating surfaces. Secondly, this scheme is augmented by a filtering method for higher order surface derivatives to improve both the stability of the extraction of feature lines and the smoothness of their appearance. [pdf]
	FreeLence - Coding with free Valences Felix Kälberer, Konrad Polthier, Ulrich Reitebuch, Max Wardetzky, Eurographics (Computer Graphics Forum), 2005, pp. 469-478. Abstract: We introduce FreeLence, a novel and simple single-rate compression coder for triangle manifold meshes. Our method uses free valences and exploits geometric information for connectivity encoding. Furthermore, we introduce a novel linear prediction scheme for geometry compression of 3D meshes. Together, these approaches yield a significant entropy reduction for mesh encoding with an average of 20-30% over leading single-rate region-growing coders, both for connectivity and geometry. [pdf]

Thesis:

Discrete Differential Operators on Polyhedral Surfaces - Convergence and Approximation, Max Wardetzky, Dissertation, Freie Universität Berlin, 2006. [dissertation online]

Mathetainment:

Kepler, oranges and shadows from the fourth dimension. ... a site for mathematical entertainment about densest ball packings, disk packings, and their connection to shadows from the 4th dimension. Prepared for the "Beliner Tag der Mathematik 2005" - a day for mathematically interested high-school students.