Discrete Differential Geometry: An Applied Introduction

Full Conference  One-Day Full Conference

Friday, 12 December, 08:30 - 17:30
Room 311
Level: Advanced

This new and elegant area of mathematics has exciting applications, as this course demonstrates by presenting practical examples in geometry processing (surface fairing, parameterization, and remeshing) and simulation (of cloth, shells, rods, and fluids).

The behavior of physical systems is typically described by a set of continuous equations using tools such as geometric mechanics and differential geometry to analyze and capture their properties. For purposes of computation, one must derive discrete (in space and time) representations of the underlying equations. Researchers in a variety of areas have discovered that theories, which are discrete from the start and have key geometric properties built into their discrete description, can often more readily yield robust numerical simulations that are true to the underlying continuous systems: they exactly preserve invariants of the continuous systems in the discrete computational realm.

This course introduces the nascent field of discrete differential geometry, laying out fundamental concepts and surveying the exciting array of applications. It begins with a simple-to-follow presentation of discrete curves and discrete curvature. This backdrop introduces the overarching theme structure of preservation, which makes repeated appearances throughout the entire course. As the day proceeds, the course explores the question of which quantities one should measure on a discrete object such as a triangle mesh, and how one should define such measurements.

Following the introduction of the basic technical concepts, the course proceeds to investigate numerous exciting application areas. The lectures introduce and delve deeply into geometric modeling problems (including variational remeshing and parameterization using discrete exterior calculus) and physical simulation of curves (such as elastic rods and hair), surfaces (such as cloth and thin-shells), and volumes (such as fluids). The emphasis is on understanding how structure preservation leads to simple and highly efficient implementations of important physical simulations.

A working knowledge of vector calculus and elementary linear algebra. Optional prerequisites: some lectures may also assume some familiarity with physical simulation, geometry processing, and triangle and tetrahedral meshes. Recommended but not required: a basic understanding of continuous local differential geometry and classical mechanics.

Intended Audience
Graduate students, researchers, and application developers who seek a unified understanding of the mathematics underlying common geometry-processing operations and how these fundamentals apply to problems such as Laplacian smoothing, surface fairing using prescribed curvature flow, remeshing, conformal parameterization, and cloth/shell/rod/fluid simulation.

Mathieu Desbrun
California Institute of Technology

Peter Schröder
California Institute of Technology

Max Wardetzky
Georg-August-Universität Göttingen

Instructor Bios

Mathieu Desbrun
Mathieu Desbrun is an associate professor at the California Institute of Technology. After receiving his PhD from the National Polytechnic Institute of Grenoble, he spent two years as a post-doctoral researcher at Caltech, and then four years as an assistant professor at the University of Southern California. His current areas of interest include: simulation and animation, with an emphasis on real-time performances (implicit integration, multires techniques); discrete surface algorithms (differential geometry, optimization, subdivision schemes, compression); implicit surfaces and level-set methods.

Peter Schröder
Peter Schröder is Professor of Computer Science and Applied and Computational Mathematics at the California Institute of Technology, where he began his academic career in 1995. Prior to Caltech and a short stint as a postdoctoral research fellow at Interval Corporation (summer 1995), he was a postdoctoral research fellow at the University of South Carolina department of mathematics and a lecturer in the computer science department, where he worked with Prof. Bjorn Jawerth and Dr. Wim Sweldens. He received his PhD in computer science from Princeton University in 1994 for work on Wavelet Methods for Illumination Computations.

Max Wardetzky
Max Wardetzky is a new assistant professor at Georg-August-Universität Göttingen. He received his PhD in mathematics from the Freie Universität Berlin in 2006, where he worked as a post-doctoral researcher until 2008. His research interests include discrete differential geometry and their applications to geometric modeling for physical simulations.