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DFG Research Center Matheon “Mathematics for key technologies”
Project F1: Discrete Differential Geometry

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Matheon Miniworkshop
Mathematical topics in surface modelling

TU-Berlin, December 15-16 2004

The workshop treats selected mathematical problems in surface modeling. The main topics are discrete differential geometry, the numerics of Riemann surfaces, and elasticity of discrete surfaces.

The main speakers will each give a talk on one of these subjects.

In addition to the three main lectures, there will also be shorter invited talks.

Main speakers

Bernard Deconinck (University of Washington)
Eitan Grinspun (Columbia University / Courant Institute)
Peter Schröder (Caltech)

Organizers

Alexander I. Bobenko
Günter M. Ziegler
Contact: Boris Springborn <springb@math.tu-berlin.de>

Location

Room MA 313/314 of the Mathematics Building
Technische Universität Berlin
Strasse des 17. Juni 136
10623 Berlin (Charlottenburg)

Schedule

Wednesday 15 December

10:00 Opening
10:15 - 11:45 Peter Schröder Discrete conformal mapping through circle packing
Lunch
13:15 - 14:45 Eitan Grinspun Adaptive solvers for thin shells: Achievements and challenges
15:15 - 15:45 Boris Springborn Variational description of circle patterns on a sphere
16:00 - 16:30 Eike Preuss Curve optimization under spatial constraints
16:45 - 17:15 Felix Kälberer Triangle mesh compression

Thursday 16 December

10:15 - 11:45 Bernard Deconinck Calculus on Riemann surfaces starting from algebraic curves
Lunch
13:15 - 13:55 Markus Schmies Computing helicoids with handles using Schottky uniformization
14:15 - 14:55 Ulrich Pinkall Design of cylinders with constant curvature
15:15 - 15:45 Tim Hoffmann &
Ulrich Pinkall
Oorange and jReality: Real time programming and experiments in virtual reality
16:15 - 17:00 Steffen Weissmann PORTAL demo

Abstracts

Bernard Deconinck

Calculus on Riemann surfaces starting from algebraic curves

Almost all integrable equations have solutions that are expressed in terms of Riemann's theta function. The well-known soliton solutions are special cases of these. The theta function solutions are parametrized in a highly transcendental way by Riemann surfaces. The nature of this parametrization has long precluded the use of these solutions in applications. In this talk, I will give a status report of an ongoing research program to make the calculation of these theta function solutions effective. Our starting point is the representation of a Riemann surface as the desingularization and compactification of a plane algebraic curve. I will describe algorithms for the compution of the monodromy of this curve, the (co)homology of the Riemann surface, and its period matrix, the computation of the theta function, and finally the computation of the Abel transform on the Riemann surface. The talk will conclude with a maple demo of implementations of all these algorithms.

Eitan Grinspun

Adaptive solvers for thin shells: Achievements and challenges

The mechanics of thin, flexible "shells" are key to physically-based surface modeling, where modeling is posed as a variational problem and the modeled surface minimizes a physical energy functional. Our interest lies in models and solvers for thin shell mechanics. Traditionally, practitioners have relied on finite-element and more recently subdivision-element formulations. Both approaches have well-understood convergence properties and are amenable to treatment by adaptive solvers---convergence and adaptivity are key components of an efficient, accurate, and predictable modeling framework. We will examine a successful and general approach to adaptivity in this context. Unfortunately, the traditional treatments of thin-shells require complex computational machinery thus prohibitive implementation cost, motivating the formulation of simple discrete models for thin shells. Such "discrete shells" are being rapidly adopted by the computer graphics community for their visually convincing results. However, the meshing-dependence of discrete shells is not yet well-understood. Furthermore, since establishing convergence is a prerequisite for building an adaptive solver, this promising approach cannot yet present itself in full bloom. We will discuss some initial results in studying convergence, and some of the current incompatibilities between constructions that ensure convergence and those that maintain discrete surface invariants.

Peter Schröder

Discrete conformal mapping through circle packing

Efficient computational procedures to construct mappings from (regions of) surfaces to the plane are essential for many applications in geometry processing and beyond. Among such mappings, conformal mappings stand out as they locally preserve angles, one of the more important distortion measures. Standard finite element formulations of the associated PDE problems are well understood, though they generally loose the underlying symmetries as they discretize a continuous notion of conformality. Instead one can approach this problem directly in the discrete setting by defining an appropriate notion of discrete conformality. This is were circle packings enter. Given a simplicial 2-manifold (with boundary) mesh, construct a mapping to the unit disk which preserves certain angles exactly and solves for the only relevant degrees of freedom, discrete area factors, directly. With the recent formulation of circle packings as solutions which minimize a variational energy very efficient numerical algorithms can be brought to bear on this problem and in my talk I will cover some of the progress we have been able to make in this realm.

Joint work with Liliya Kharevych and Boris Springborn.


Boris Springborn . 14.12.2004. Accesses: [an error occurred while processing this directive] since 11.03

DFG Research Center Matheon "Mathematics for key technologies"