Discrete Differential Geometry

Geometry Processing and Physical Simulations

Monday June 18, 2012, Chapel Hill, North Carolina, USA

Please register from the SoCG registration website.


Special workshop co-located with ACM Symposium on Computational Geometry

Schedule Monday June 18, 2012

2:20-3:00 Eitan Grinspun (Columbia): A Geometric Approach to Computation of Elastica in Contact
3:00-3:30 Pooran Memari (CNRS-Telecom ParisTech): Hodge-Optimized Triangulations
3:30-4:00 Keenan Crane (Caltech): Curves without Stretching, Surfaces without Shearing: Manipulating Geometry via Extrinsic Curvature

Break

4:20-4:50 Fernando de Goes (Caltech): Blue noise through Power Diagrams
4:50-5:20 Etienne Vouga (Columbia): Geometry of Self-supporting Surfaces
5:20-6:00 Yiying Tong (Michigan State):Discrete Fundamental Forms on Surfaces

Abstracts

Description

Discrete Differential Geometry is an emerging field at the intersection of differential geometry, computer graphics, applied mathematics, and computational physics.

Geometry, in its most general form, is the study of space and of the properties of shapes in space. Ever since Euclid, geometry has been at the foundation of the most successful physical models and theories (from quantum dynamics to general relativity) as it provides a concise way to mathematically formalize apparent symmetries and experimental invariants. Since Newton and Leibniz, such geometric modeling has mostly relied on a deeply-rooted smooth (i.e., differential) comprehension of the world. However, this abstraction of differentiability inherently clashes with a computer's ability of storing only finite sets of numbers.

In recent years, the interplay of differential geometry, computational mathematics, and computational geometry has brought new insights into the mathematics of surface and volume meshes and physical simulations using such meshes. This has come about through the study of Discrete Differential Geometry by investigating how to process, analyze, edit, and animate discrete geometric data such as meshes, graphs, and point sets, and how to discretize differential operators for use in physical simulations.

Starting with discrete data from the very beginning offers a very different, yet complementary perspective on geometry. While still a young field, Discrete Differential Geometry has already led to novel solutions to demanding engineering applications in the form of innovative, efficient algorithms for geometry processing and modeling, often with improved numerics when compared to traditional numerical methods. Interesting links between the discrete, computational realm and its continuous counterpart have surfaced. Although Discrete Differential Geometry has mainly focused on Euclidean, spherical and hyperbolic geometry, we are starting to see incursions into more general type of geometries, such as conformal and Moebius geometry (for parameterization, for instance), and even symplectic geometry (in the context of discrete mechanics).

Discrete Differential Geometry is a quite diverse field, due to the ubiquity of geometry in scientific modeling. Usual topics include discrete differential geometry of curves, surfaces, and volumes; discrete versus continuous curvature and other differential operators; topological aspects of simplicial and polygonal meshes; (re)meshing; interaction between mesh quality and discrete operators; multiresolution and hierarchical representations; animation and simulation. Publications in this field are generally found in conference publications such as ACM/EG Symposium on Geometry Processing, ACM SIGGRAPH, and Eurographics, as well as many other Computer Graphics and Mathematics journals including SIAM journals.

This special workshop is intended to provide an opportunity for interaction between computational geometry researchers and researchers in Discrete Differential Geometry.

Organizers

Anil N. Hirani, University of Illinois at Urbana-Champaign
Mathieu Desbrun, California Institute of Technology