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Selected Topics in Discrete Differential Geometry and Visualization,
Peking University, 2009
Content
Overview
This course is part of the
"National Mathematics Graduate Summer School"
(
Google translation)
that takes place at
Peking University
from July 12 to August 8. See also the
Chinese homepage
of this course. This course is supported by DFGProject Geometric Problems and Special PDEs.
Surfaces and curves in space are of central importance in many
application areas like Computer Graphics, Physics Simulation or
Architecture. While Differential Geometry is concerned with smooth
surfaces and curves, within a computer they are always represented as
a finite set of points, connected by triangles or line segments. This
means that surfaces are really treated as being polyhedral and curves
are treated as polygons.
Discrete Differential Geometry is a very active area of research where
(instead of looking at discrete objects just as numerical
approximations to the smooth ones) the goal is to develop a theory of
discrete curves and surfaces that has the same structure as the
corresponding smooth theory. Quite often this approach leads to
solutions that are "exact" on the discrete level (not just
approximations) and provide highly efficient new algorithms.
Topics

Discrete surfaces with constant negative curvature
 Ulrich Pinkall.
Designing Cylinders with Constant Negative Curvature.
In Discrete Differential Geometry, pages 5766. Springer 2008.
(download PDF)

Alexander Bobenko and Ulrich Pinkall.
Discrete surfaces with constant negative Gaussian
curvature and the Hirota equation.
J. Differential Geom., 43(3):527611, 1996.
(online)

Alexander I. Bobenko and Ulrich Pinkall.
Discretization of surfaces and integrable systems.
In Discrete integrable geometry and physics (Vienna, 1996),
pages 358. Oxford Univ. Press, New York, 1999.
(download PDF)


Discrete vortex lines in fluids
U. Pinkall, B. Springborn, S. Weißmann.
A new doubly discrete analogue of smoke ring flow and the real
time simulation of fluid flow
J. Phys. A: Math. Theor. 40 (2007), 1256312576.
(online,
arXiv:0708.0979 [nlin.SI])


Elastic deformations of discrete surfaces


Conformal maps between discrete surfaces
B. Springborn, P. Schröder, U. Pinkall.
Conformal Equivalence of Triangle Meshes.
ACM Transactions on Graphics 27:3 [Proceedings of ACM SIGGRAPH 2008]
(online,
download PDF)

Praxis Sessions
The lectures will be accompanied by practical introduction to using
and developing software in this field.
Ready made examples
The following examples provide ready to use applications to experiment
with. No programming knowledge is required.
More Math Labs at the
website of the geometry group at TUBerlin...
Basic tutorials
Follwing the steps below will enable someone with little programming
knowledge to do some math with jReality.
Session 1

Set up jReality as an eclipse project:
Follow the description on the jreality wiki
(or download
the eclipse folder and
download the prepared
workspace,
unzip them and start eclipse  look for eclipse.exe. When eclipse
asks for a workspace use the
downloaded workspace and add 2 general projects "jreality" and
"ddgvis": got the menu File>New>Project...>General>Project,
press "Next" and then use the exact names of the projects, press "Finish").
 Learn how to Display a geometry.

Study the tutorial on how to
use a parametric surface factory.

Change the tutorial example in order to visualize a surface of
your choice.

Display the surface
in JRViewerVR
Session 2

Add an inspector panel for the ParametricSurfaceFactory in the
example above. For this you need to download the new
class ViewShrinkPanelPlugin.java
and copy it to your eclipse project.

Choose a one parameter family of surfaces (e.g. a minimal surface
with its associated family) and implement it.
Add
a slider for this parameter.

Have a look at the "Navigator" (you will find it in the right slot
which you may open in the "Window" menu or with CtrlShiftRight).

Go through the
introduction tutorials
of the
jReality developer tutorial. You may find the source code
of these tutorials in the eclipse project "jreality" in the
folder "srctutorial" in the package "de.jreality.tutorial.intro".
Session 35: Advanced tutorial
In the Sessions 35 you should do a small project of your own,
probably in groups of 2 students. Here are 4 suggestions for projects,
that apply the theory taught in the course.
To get started study
the
discrete Ksurfaces tutorial
or the
Polygon
editor tutorial and extend them.
From right to left: Yucheng Lu, Haoshu Tian, Steffen Weissmann,
Ulrich Pinkall, Paul Peters, Ruru Hao, Wei Jin, Wei Huayi, Pan Hao,
Jingyang Zhong
Ulrich
Pinkall
G. Paul
Peters
Steffen
Weißmann
