I have the pleasure of heading the Discrete Differential Geometry Lab at the University of Goettingen, with its wonderful people.
Discrete Differential Geometry, Geometry Processing, Computational Topology, Physical Simulation, Computer Graphics.

Institute of Num. and Appl. Math University of Göttingen Lotzestr. 1618 37083 Göttingen, Germany 
Vox: +49 (0)551 3922235 Fax: +49 (0)551 3933944 Net: lastname (at) math.unigoettingen.de 

Persistent Homology meets Statistical Inference  A Case Study: Detecting Modes of OneDimensional Signals Ulrich Bauer, Axel Munk, Hannes Sieling, and Max Wardetzky. Abstract: We investigate the problem of estimating persistent homology of noisy one dimensional signals. We relate this to the problem of estimating the number of modes (i.e., local maxima)  a well known question in statistical inference  and we show how to do so without presmoothing the data. To this end, we extend the ideas of persistent homology by working with norms different from the (classical) supremum norm. As a particular case we investigate the so called Kolmogorov norm. We argue that this extension has certain statistical advantages. We offer confidence bands for the attendant Kolmogorov signatures, thereby allowing for the selection of relevant signatures with a statistically controllable error. As a result of independent interest, we show that socalled taut strings minimize the number of critical points for a very general class of functions. We illustrate our results by several numerical examples. [pdf] 

Wire Mesh Design Akash Garg, Andrew SagemanFurnas, Bailin Deng, Yonghao Yue, Eitan Grinspun, Mark Pauly, and Max Wardetzky. ACM Transactions on Graphics 33:4, pp. 66:1–66:12, 2014. Abstract: We present a computational approach for designing wire meshes, i.e., freeform surfaces composed of woven wires arranged in a regular grid. To facilitate shape exploration, we map material properties of wire meshes to the geometric model of Chebyshev nets. This abstraction is exploited to build an efficient optimization scheme. While the theory of Chebyshev nets suggests a highly constrained design space, we show that allowing controlled deviations from the underlying surface provides a rich shape space for design exploration. Our algorithm balances globally coupled material constraints with aesthetic and geometric design objectives that can be specified by the user in an interactive design session. In addition to sculptural art, wire meshes represent an innovative medium for industrial applications including composite materials and architectural façades. We demonstrate the effectiveness of our approach using a variety of digital and physical prototypes with a level of shape complexity unobtainable using previous methods. [lowres pdf] [highres pdf] [Video] 

Exploring the Geometry of the Space of Shells Behrend Heeren, Martin Rumpf, Peter Schröder, Max Wardetzky, and Benedikt Wirth. Computer Graphics Forum 33(5):247–256, 2014. Abstract: We prove both in the smooth and discrete setting that the Hessian of an elastic deformation energy results in a proper Riemannian metric on the space of shells (modulo rigid body motions). Based on this foundation we develop a time and spacediscrete geodesic calculus. In particular we show how to shoot geodesics with prescribed initial data, and we give a construction for parallel transport in shell space. This enables, for example, natural extrapolation of paths in shell space and transfer of large nonlinear deformations from one shell to another with applications in animation, geometric, and physical modeling. Finally, we examine some aspects of curvature on shell space. [pdf] 

Functional Fluids on Surfaces Omri Azencot, Steffen Weißmann, Maks Ovsjanikov, Max Wardetzky, and Mirela BenChen. Computer Graphics Forum 33(5):237–246, 2014. Abstract: Fluid simulation plays a key role in various domains of science including computer graphics. While most existing work addresses fluids on bounded Euclidean domains, we consider the problem of simulating the behavior of an incompressible fluid on a curved surface represented as an unstructured triangle mesh. Unlike the commonly used Eulerian description of the fluid using its timevarying velocity field, we propose to model fluids using their vorticity, i.e., by a (time varying) scalar function on the surface. During each time step, we advance scalar vorticity along two consecutive, stationary velocity fields. This approach leads to a variational integrator in the space continuous setting. In addition, using this approach, the update rule amounts to manipulating functions on the surface using linear operators, which can be discretized efficiently using the recently introduced functional approach to vector fields. Combining these time and space discretizations leads to a conceptually and algorithmically simple approach, which is efficient, timereversible and conserves vorticity by construction. We further demonstrate that our method exhibits no numerical dissipation and is able to reproduce intricate phenomena such as vortex shedding from boundaries. [pdf] [Video] 

Geodesics in Heat: A New Approach to Computing Distance Based on Heat Flow Keenan Crane, Clarisse Weischedel, and Max Wardetzky. ACM Transaction on Graphics 32:5, pp. 152:1152:11, 2013. Abstract: We introduce the heat method for computing the geodesic distance to a specified subset (e.g., point or curve) of a given domain. The heat method is robust, efficient, and simple to implement since it is based on solving a pair of standard linear elliptic problems. The resulting systems can be prefactored once and subsequently solved in nearlinear time. In practice, distance is updated an order of magnitude faster than with stateoftheart methods, while maintaining a comparable level of accuracy. The method requires only standard differential operators and can hence be applied on a wide variety of domains (grids, triangle meshes, point clouds, etc.). We provide numerical evidence that the method converges to the exact distance in the limit of refinement; we also explore smoothed approximations of distance suitable for applications where greater regularity is required. [pdf] 

A discrete geometric approach for simulating the dynamics of thin viscous threads Basile Audoly, Nicolas Clauvelin, PierreThomas Brun, Miklos Bergou, Eitan Grinspun, Max Wardetzky. Journal of Computational Physics, volume 253, pp. 18–49, 2013. Abstract: We present a numerical model for the dynamics of thin viscous threads based on a discrete, Lagrangian formulation of the smooth equations. The model makes use of a condensed set of coordinates, called the centerline/spin representation: the kinematic constraints linking the centerline?s tangent to the orientation of the material frame is used to eliminate two out of three degrees of freedom associated with rotations. Based on a description of twist inspired from discrete differential geometry and from variational principles, we build a fullfledged discrete viscous thread model, which includes in particular a discrete representation of the internal viscous stress. Consistency of the discrete model with the classical, smooth equations for thin threads is established formally. Our numerical method is validated against reference solutions for steady coiling. The method makes it possible to simulate the unsteady behavior of thin viscous threads in a robust and efficient way, including the combined effects of inertia, stretching, bending, twisting, large rotations and surface tension. [pdf] 

TimeDiscrete Geodesics in the Space of Shells Behrend Heeren, Martin Rumpf, Max Wardetzky and Benedikt Wirth. Computer Graphics Forum 31(5):1755–1764, 2012. Abstract: Building on concepts from continuum mechanics, we offer a computational model for geodesics in the space of thin shells, with a metric that reflects viscous dissipation required to physically deform a thin shell. Different from previous work, we incorporate bending contributions into our deformation energy on top of membrane distortion terms in order to obtain a physically sound notion of distance between shells, which does not require additional smoothing. Our bending energy formulation depends on the socalled relative Weingarten map, for which we provide a discrete analogue based on principles of discrete differential geometry. Our computational results emphasize the strong impact of physical parameters on the evolution of a shell shape along a geodesic path. [pdf] 

Flexible Developable Surfaces Justin Solomon, Etienne Vouga, Max Wardetzky, Eitan Grinspun. Computer Graphics Forum 31(5):1567–1576, 2012. Abstract: We introduce a discrete paradigm for developable surface modeling. Unlike previous attempts at interactive developable surface modeling, our system is able to enforce exact developability at every step, ensuring that users do not inadvertently suggest configurations that leave the manifold of admissible folds of a flat twodimensional sheet. With methods for navigation of this highly nonlinear constraint space in place, we show how to formulate a discrete mean curvature bending energy measuring how far a given discrete developable surface is from being flat. This energy enables relaxation of usergenerated configurations and suggests a straightforward subdivision scheme that produces admissible smoothed versions of bent regions of our discrete developable surfaces. [pdf] 

Optimal topological simplification of discrete functions on surfaces Ulrich Bauer, Carsten Lange, and Max Wardetzky. Discrete and Computational Geometry 47:2 (2012), 347–377. Abstract: Given a function f on a surface and a tolerance δ > 0, we construct a function f_{δ} subject to ‖f_{δ}  f‖_{∞} ≤ δ such that f_{δ} has a minimum number of critical points. Our construction relies on a connection between discrete Morse theory and persistent homology and completely removes homological noise with persistence ≤ 2δ from the input function f. The number of critical points of the resulting simplified function f_{δ} achieves the lower bound dictated by the stability theorem of persistent homology. We show that the simplified function can be computed in linear time after persistence pairs have been computed. [pdf] [doi] 

Discrete Laplacians on General Polygonal Meshes Marc Alexa and Max Wardetzky, ACM Transaction on Graphics 30:4 (SIGGRAPH), pp. 102:1102:10, 2011. Abstract: While the theory and applications of discrete Laplacians on triangulated surfaces are well developed, far less is known about the general polygonal case. We present here a principled approach for constructing geometric discrete Laplacians on surfaces with arbitrary polygonal faces, encompassing nonplanar and nonconvex polygons. Our construction is guided by closely mimicking structural properties of the smooth LaplaceBeltrami operator. Among other features, our construction leads to an extension of the widely employed cotan formula from triangles to polygons. Besides carefully laying out theoretical aspects, we demonstrate the versatility of our approach for a variety of geometry processing applications, embarking on situations that would have been more difficult to achieve based on geometric Laplacians for simplicial meshes or purely combinatorial Laplacians for general meshes. [pdf] [code] 

Total Variation Meets Topological Persistence: A First Encounter Ulrich Bauer, CarolaBibiane Schönlieb, and Max Wardetzky. Proceedings of ICNAAM 2010, pp. 10221026. Abstract: We present first insights into the relation between two popular yet apparently dissimilar approaches to denoising of one dimensional signals, based on (i) total variation (TV) minimization and (ii) ideas from topological persistence. While a close relation between (i) and (ii) might phenomenologically not be unexpected, our work appears to be the first to make this connection precise for one dimensional signals. We provide a link between (i) and (ii) that builds on the equivalence between TVL2 regularization and taut strings and leads to a novel and efficient denoising algorithm that is contrast preserving and operates in O(nlogn) time, where n is the size of the input. [pdf] 

Discrete Viscous Threads Miklos Bergou, Basile Audoly, Etienne Vouga, Max Wardetzky, Eitan Grinspun, ACM Transaction on Graphics 29:4 (SIGGRAPH), pp. 116:1116:10, 2010. Abstract: We present a continuumbased discrete model for thin threads of viscous fluid by drawing upon the Rayleigh analogy to elastic rods, demonstrating canonical coiling, folding, and breakup in dynamic simulations. Our derivation emphasizes spacetime symmetry, which sheds light on the role of timeparallel transport in eliminating  without approximation  all but an O(n) band of entries of the physical system's energy Hessian. The result is a fast, unified, implicit treatment of viscous threads and elastic rods that closely reproduces a variety of fascinating physical phenomena, including hysteretic transitions between coiling regimes, competition between surface tension and gravity, and the first numerical fluidmechanical sewing machine. The novel implicit treatment also yields an order of magnitude speedup in our elastic rod dynamics. [pdf] [Video] 

Uniform Convergence of Discrete Curvatures from Nets of Curvature Lines Ulrich Bauer, Konrad Polthier, Max Wardetzky, Discrete and Computational Geometry 43:4, 798–823, 2010. Abstract: We study “Steinertype” 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. [pdf] [doi] 

Discrete Elastic Rods Miklos Bergou, Max Wardetzky, Stephen Robinson, Basile Audoly, Eitan Grinspun, ACM Transaction on Graphics 27:3 (SIGGRAPH), pp. 63:163:12, 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 rigidbodies 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 coupledmode experiments, and via qualitative knottying 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 3space. In particular, we show that mean curvature vectors converge in the sense of distributions, but fail to converge in L^2. [pdf] 

TRACKS: Toward Directable Thin Shells Miklos Bergou, Saurabh Mathur, Max Wardetzky, Eitan Grinspun, ACM Transaction on Graphics 26:3 (SIGGRAPH), pp. 50:150:10, 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 weakform 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] 

Discrete Quadratic Curvature Energies Max Wardetzky, Miklos Bergou, David Harmon, Denis Zorin, Eitan Grinspun, Computer Aided Geometric Design (CAGD) 24, 2007, pp. 499518. Abstract: We present a family of discrete isometric bending models (IBMs) for triangulated surfaces in 3space. 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 timeintegration of cloth dynamics with a two to threefold net speedup over existing nonlinear methods, and nearinteractive rates for Willmore smoothing of large meshes. [pdf] [Video] 

Discrete Laplace operators: No free lunch Max Wardetzky, Saurabh Mathur, Felix Kälberer, Eitan Grinspun, Symposium on Geometry Processing, 2007, pp. 3337. 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 wellknown and widelyused operators. [pdf] 

Cubic Shells Akah Garg, Eitan Grinspun, Max Wardetzky, Denis Zorin, Symposium on Computer Animation, 2007, pp. 9198. Abstract: Hingebased bending models are widely used in the physicallybased animation of cloth, thin plates and shells. We propose a hingebased 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] 

On the Convergence of Metric and Geometric Properties of Polyhedral Surfaces Klaus Hildebrandt, Konrad Polthier, Max Wardetzky, in Geometriae Dedicata 123, 2006, pp. 89112. Abstract: We provide conditions for convergence of polyhedral surfaces and their discrete geometric properties to smooth surfaces embedded in Euclidean 3space. 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 LaplaceBeltrami 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. 227230. Abstract: Efficient computation of curvaturebased 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 curvaturebased 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] 

FreeLence  Coding with free Valences
Felix Kälberer, Konrad Polthier, Ulrich Reitebuch, Max Wardetzky, Eurographics (Computer Graphics Forum), 2005, pp. 469478. Abstract: We introduce FreeLence, a novel and simple singlerate 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 2030% over leading singlerate regiongrowing coders, both for connectivity and geometry. [pdf] 

Smooth Feature Lines on Surface Meshes Klaus Hildebrandt, Konrad Polthier, Max Wardetzky, Symposium on Geometry Processing, 2005, pp. 8590. 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] 

Convergence of Discrete Elastica Henrik Schumacher, Sebastian Scholtes, Max Wardetzky, Oberwolfach Reports No. 34/2012. Abstract: Using techniques related to the notions of epigraph distance and AttouchWetsconvergence, we show that under appropriate boundary conditions discrete elastica (i.e., polygonal curves of some fixed length that minimize a certain discrete bending energy) converge to smooth elastica (i.e., smooth curves of some given length that minimize smooth bending energy). [pdf] 

Persistence Simplifiation of Discrete Morse Functions on Surfaces Ulrich Bauer, Carsten Lange, Max Wardetzky, Oberwolfach Reports, Volume 6, Issue 1, 2009. Abstract: We combine the concept of persistent homology with Forman's discrete Morse theory on regular 2manifold CW complexes to solve the problem of minimizing the number of critical points among all functions within a prescribed distance from a given input function. We give a constructive proof of the tightness of the lower bound on the number of critical points provided by the Stability Theorem of persistent homology. [pdf] 

Geometric Aspects of Discrete Elastic Rods Max Wardetzky, Miklos Bergou, Stephen Robinson, Basile Audoly, Eitan Grinspun, Oberwolfach Reports, Volume 6, Issue 1, 2009. Abstract: Elastic rods are curvelike elastic bodies that have one dimension (length) much larger than the others (crosssection). Their elastic energy breaks down into three contributions: stretching, bending, and twisting. Stretching and bending are captured by the deformation of a space curve called the centerline, while twisting is captured by the rotation of a material frame associated to each point on the centerline. Building on the notions of framed curves, parallel transport, and holonomy, we present a smooth and a corresponding discrete theory that establishes an efficient model for simulating thin flexible rods with arbitrary cross section and undeformed configuration. [pdf] 

Algebraic Topology on Polyhedral Surfaces from Finite Elements Max Wardetzky, Klaus Hildebrandt, Konrad Polthier, Oberwolfach Reports 12/2006. Abstract: We report on a development using piecewise constant vector fields (or oneforms) 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. [pdf] 

SIGGRAPH ASIA 2008 COURSE NOTES on Discrete Differential Geometry Course Organizors: Eitan Grinspun and Max Wardetzky Abstract: This 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. [pdf] 

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

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 highschool students. 

Blind Dates für die Wissenschaft Brigitte LutzWestphal und Max Wardetzky, Mitteilungen der DMV, 2008. Abstract: Wissenschaftlicher 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 DFGForschungszentrum MATHEON entwickelten wir ein neues Konzept, um einen vertieften wissenschaftlichen Austausch besser zu fördern. [pdf] 