Table Of ContentVisualization and Mathematics
Springer-V erlag Berlin Heidelberg GmbH
Rans-Christian Rege
Konrad Polthier (Eds.)
Visualization
and Mathematies
Experiments, Simulations
and Environments
With 187 Figures, 43 in Color
Springer
Hans-Christian Hege
Wissenschaftliche Visualisierung
Konrad-Zuse-Zentrum fiir Informationstechnik Berlin
TakustraBe 7
D-14195 Berlin, Germany
e-mail: [email protected]
Konrad Polthier
Fachbereich 3, Mathematik
Technische Universităt Berlin
StraBe des 17. Juni 136
D-10623 Berlin, Germany
e-mail: [email protected]
Cataloging-in-Publication Data applied for.
Die Deutsche Bibliothek -CIP-Einheitsaufnahme
Visualization and mathematics: experiments, simulations and environments 1 Hans-Christian
Hege; Konrad Polthier (ed.). - Berlin; Heidelberg; New York; Barcelona; Budapest; Hong Kong;
London; Milan; Paris; Santa Clara; Singapore; Tokyo: Springer, 1997
ISBN 978-3-642-63891-6 ISBN 978-3-642-59195-2 (eBook)
DOI 10.1007/978-3-642-59195-2
Mathematics Subject Classification (1991): 65-04,53-04, 65S05, 68U05
ISBN 978-3-642-63891-6
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Preface
Human perception and communicationaboutcomplexsubjectsisstronglyfa
cilitated by images and visual models. Computer techniques have extremely
simplified the visualization process, and visualization has become an invalua
ble tool in scientific research, even in abstract disciplines like mathematics.
On the other hand visualization techniques are based on mathematical con
cepts and methods. This book provides a collection of research articles on
the rapidly evolving, interdisciplinary areaofvisualization and mathematics.
In mathematics, young disciplines like mathematical visualization and
experimentalmathematics utilizecomputersandtheirgraphicscapabilitiesto
solveintricatemathematical problems. Due to thespecific natureofproblems
from the different branches of applied and pure mathematics, sophisticated
visualization methods complementing numerical and symbolical algorithms
are required to make the computer applicable as an experimental device for
mathematicians.
In computer graphics, the use of mathematical methods has a long hi
storyand the influenceofmathematicsstillincreases. Manycomputergraphi
cal methods and visualization techniques require fairly elaborate theoretical
concepts and mathematical algorithms.
Insummary,bothcommunitiesdevelopnewtechniques that areofinterest
for both areas. Cooperationbetweenthe correspondingresearch communities
thereforeisanimportantissue.Ithasalreadyled tomanymutualbenefitsand
has inspirednumerousdevelopmentsin applications that exploitvisualization
techniques.
This book grew out of the international workshop "Visualization and
Mathematics",held from May 30to June 2, 1995 in Berlin-Dahlem with sup
port of the Deutsche Forschungsgemeinschaft (DFG) and the Max-Planck
Gesellschaft (MPG). The aim ofthis workshop was to bring together mathe
maticians, computer graphics experts, and professionals in scientific visuali
zation in order to report on latest research and discuss new developments.
The selections represent a broad spectrum of new techniques being used in
mathematics and computergraphics. The bookisoneofthefirst wherelatest
research material is collected around the common interests of both commu
nities.
The chapter VISUALIZING MATHEMATICS focuses on visualization que
stions of special mathematical phenomena. The long history of sphere ever
sions is continued by G. Francis, 1.Sullivan, R.Kusner, K. Brakke, C. Hart
mann, and G. Chapel with a new approach where the eversion is driven by
the gradient flow ofan elastic bending energy. They also report about using
the Cave environment for demonstrating their simulations. B. Hausmann,
B. Slopianka, and H.P. Seideldemonstrate experimental and interactive soft-
VI Preface
warefor exploringthe characteristic propertiesofplane hyperbolicgeometry.
G. Martin and I. Sterlingapply equivariant geometry and use visualization in
order to study non-linear electrodynamics. The knowledge and classification
of singularities of maps is an important question in singularity theory and
requests special visualization techniques. In this context R. Morris reports
about recent experiments and presents new visualization tools. D.Roseman
studies the problem of visualizing 4-dimensional euclidean space. He discus
ses various techniques and applies them to knotted surfaces as case studies.
D. Saupe and M. Ruhldemonstrate how to study parameter-dependent alge
braic surfaces by combining several pieces of readily available software and
off-the-shelfhardware.
Experiments have a long tradition also in geometry, and recent develop
ments are the topic of the chapter GEOMETRIC ALGORITHMS AND EXPE
RIMENTS. A surprising new optimal candidate to the Lord Kelvin problem
of partitioning space into equal volume cellswhile using the least interface
area has recently been discovered by numerical experiments of Weaire and
Phelan. K. Brakke and J. Sullivan report on the use ofsymmetry features of
the softwarepackageSurfaceEvolver tostudy such foam problems. K. Grofle
Brauckmann and K. Polthier discover new examples of constant mean cur
vature surfaces using the Wente torus and Delauney surfaces as building
blocks. In their construction the numerical algorithm of B. Oberknapp and
K. Polthier, based on new techniques ofdiscrete geometry, is employed. This
way computation and visualization of a broad range of constant mean cur
vature surfaces has become possible for the first time. D. Huson discusses
algorithms and datastructures for the systematic enumeration, visualization
and interactive manipulation of periodic tHings which are the base of his
visualization tool FunTiles.
VISUALIZATION ALGORITHMS AND DATA STRUCTURES areofutmost im
portancefor manycomputergraphicaltasks. M. Kohlerand H. Miillerpresent
a new subdivision scheme for smoothing polygonal meshes. It uses an effi
cient depth-first technique with considerable better space complexity than
previous methods. H. Battke, D. Stallingand H.C. Hege describe how the line
integral convolution method can be generalized for visualization of vector
fields on arbitrary surfaces in 3D space. In large scale simulations various
grid types may occur and data may be distributed. M. Rumpfand B. Schupp
propose a concept for elegantly integrating visualization components in such
simulation environments by using procedural data access.
VISUALiZATION ENVIRONMENTS serve as a laboratory for scientific expe
riments and simulations. M. Alefeld, J. Haber and A.Heim present a new 3D
graphics library for scientific visualization that exploits graphics hardware
also for high-quality image synthesis. Versatile visualization environments
usually are rather complex software systems. In contrast the following two
approaches focus on minimality to simplify usability. E. Beierdescribes a ge
neric graphics kernel which can be easily customized to special applications
Preface VII
and reports about a derived kernel for use in the furniture industry. The 3D
rendering environment MRT of D. Fellner consists of customizable building
blocks that make it possible to build visualization packages and testbeds for
new algorithms rather efficiently. The visualization environment Oorange by
C. Gunn, A. Ortmann, U.Pinkall, K. Polthier and U. Schwarz is a virtual la
boratory especially designed for the needs of experimental mathematics. It
offers software features for each component of a traditional physical experi
ment. J. Lemordantdescribesaframeworkfor visualizationoftime-dependent
multidimensional datawhichisimplementedin thesystemHopi.It usesa da
taflow model and separates the data reduction process from the presentation
part. Extendable graphics toolkits are powerful tools in scientific research
and gain by combination with numerical packages. J. Walton and M. Dewar
report on the use of various toolkits (IrisExplorer, OpenInventor, Axiom)
to visualize and understand numerical data. They put special emphasis on
a common data format for 3D geometry that allows to share objects and
scenes between applications and to distribute visualizations over the World
Wide Web.
In the last part VISUALIZATION AND SIMULATION TECHNIQUES the use
ofcomputer graphics and mathematicalmethods in various applicationfields
are discussed. R. Beck, P. Deufihard, H.C. Hege, M. Seebafi and D. Stalling
present numerical and visualization aspects of medical treatment planning
and describe the framework Hyperplan that supports typical working steps
in therapy planning. R. Malladi and J. Sethian introduce level set methods,
powerfulnumerical toolsfor trackingtheevolutionofinterfacesmovingunder
a variety offorces. Beside a review of numerical and geometric results they
introduce new schemes for recovery of 3D shapes and image enhancement.
Treatment of compressible flows in 2D and 3D requires sophisticated tools
from numerics and visualization. The report of M. Wierse, T. Gefiner and
D. Kroner addresses these topics by demonstrating how an industrial flow
problem is solved.
We thank all authors for their contributions and hope that the achieve
ments presentedin this bookas well as the new questionsarisingin thesequel
will lead to a continued and joint development in this exciting, interdiscipli
nary area.
Berlin, 1996 Hans-Christian Hege . Konrad Polthier
Table of Contents
Preface. ....................................................... V
Contributors XV
Part I. Visualizing Mathematics
The Minimax Sphere Eversion
GeorgeFrancis, JohnM.Sullivan, RobB.Kusner, KenA.Brakke,
ChrisHartman, and GlennChappell 3
1. A History ofSphere Eversions ................................. 3
2. The Minimax Sphere Eversion. ................................ 5
3. Using the Surface Evolver to Evert a Sphere. .................... 7
4. A Topological Description of the Eversion. ...................... 11
5. Virtual Environments and Supercomputing'95 ................... 16
Exploring Plane Hyperbolic Geometry
BarbaraHausmann, BrittaSlopianka, and Hans-PeterSeidel 21
1. Introduction................................................. 21
2. Plane Euclidean and Hyperbolic Geometry. ..................... 22
3. DifferentEuclideanModelsfor HyperbolicGeometryand Geometric
Transformations ............................................. 23
4. Hyperbolic Escher Patterns 31
5. Software Implementation Details. .............................. 33
6. Conclusion and Further Work 35
Visualizing Nonlinear Electrodynamics
Geoffrey Martin and Ivan Sterling. ................................ 37
1. Introduction................................................. 37
2. Review ofMartin Theory. .................................... 38
3. Balanced Fields for R8 .....................•................. 39
4. The First Reduction - to a Euclidean R4 . . . . . . • . . . . . . . . . . . . . . ... 46
5. Electric and Magnectic Fields 47
6. Radially Symmetric. ......................................... 49
X Table ofContents
The Use of Computer Graphics for Solving Problems in
Singularity Theory
Richard J. Morris ............................................... 53
1. Introduction................................................. 53
2. A Toolkit for Exploring Singularity Theory. ..................... 54
3. Two Problems in Singularity Theory 60
What Should a Surface in 4-Space Look Like?
DennisRoseman. ............................................... 67
1. Introduction................................................. 67
2. Mathematical Setting. ........................................ 68
3. What Mathematical Questions Do We Want to Address? ......... 69
4. How to Improve a Knotted Surface? .. ......................... 69
5. Visualization in 4-Dimensions: General 70
6. A Concrete Example .. ....................................... 72
7. Visualization Solutions in m.4: Projections, Slices, and Slabs .. ..... 72
8. Visualization Solutions in m.4 : Adding the "Missing" Information . 74
9. Some Current Visualization Tools. ............................. 78
10. Notes on the Figures. ........................................ 79
Animation ofAlgebraic Surfaces
Dietmar Saupe and Matthias Ruhl ................................ 83
1. Introduction................................................. 83
2. Raytracing Algebraic Surfaces. ................................ 85
3. Polygonalization and Physically-based Interactive Animation. ..... 88
4. Conclusion.................................................. 91
Part II. Geometric Algorithms and Experiments
Using Symmetry Features ofthe Surface Evolver to Study
Foams
KenA.Brakke and JohnM.Sullivan 95
1. Introduction................................................. 95
2. The Foams ofKelvin and Weaire-Phelan 96
3. Triply Periodic Foams in the Evolver ........................... 98
4. Using Two-fold Symmetries ofMinimal Surfaces for the
Kelvin Foam 104
5. Other Rotational Symmetries and the Weaire-Phelan Foam 108
6. Using Edge Energies to Eliminate Faces in Mirror Planes 114
Table ofContents XI
Constant Mean Curvature Surfaces Derived from Delaunay's
and Wente's Examples
KarstenGrof3e-Brauckmann and KonradPolthier 119
1. Introduction 119
2. Delaunay Unduloids and Wente Tori 121
3. Modifying the Wente Torus 122
4. Moduli Spaces ofEmbedded Symmetric Surfaces with
Delaunay Ends " 125
Visualization ofPeriodic THings
DanielH.Huson 135
1. Tilings and Patterns 135
2. A Fundamental Data-Structurefor Periodic Tilings 136
3. Visualization of2-Dimensional Tilings 137
4. Software 138
An Algorithm for Discrete Constant Mean Curvature
Surfaces
Bernd Oberknapp and Konrad Polthier 141
1. Introduction 141
2. Prerequisites 143
3. Discrete Harmonic Maps and Minimal Surfaces in §3 146
4. Discrete CMC Surfaces via Conjugation 151
5. Conjugate Surface Construction and Symmetry 155
6. Numerical Considerations and Visualization 158
Part III. Visualization Algorithms and Data Structures
Efficient Calculation ofSubdivision Surfaces for Visualization
MarkusKohler and HeinrichMuller 165
1. Introduction 165
2. Doo-Sabin Subdivision Surfaces 166
3. Depth-First Subdivision 167
4. Tracing Strategies 176
5. Examples 177
Fast Line Integral Convolution for Arbitrary Surfaces in 3D
Henrik Battke, Detlev Stalling, and Hans-Christian Hege 181
1. Introduction 181
2. Background 182
3. LIC on Surfaces 184
4. Projecting 3D-Fields 192
