Visualization and Mathematics

Transcription

1 Hans-Christian Hege Konrad Polthier (Eds.) Visualization and Mathematics Experiments, Simulations and Environments With 187 Figures, 43 in Color «P Springer

2 Table of Contents Preface Contributors V XV Part I. Visualizing Mathematics The Minimax Sphere Eversion George Francis, John M. Sulli van, RobB.Kusner, KenA.Brakke, Chris Hartman, and Glenn Chappell 3 1. A History of Sphere 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 Virtual Environments and Supercomputing'95 16 Exploring Plane Hyperbolic Geometry Barbara Hausmann, Britta Slopianka, and Hans-Peter Seidel Introduction Plane Euclidean and Hyperbolic Geometry Different Euclidean Models for Hyperbolic Geometry and Geometrie Transformations Hyperbolic Escher Patterns Software Implementation Details Conclusion and Further Work 35 Visualizing Nonlinear Electrodynamics Geoffrey Martin and Ivan Sterling Introduction Review of Martin Theory Balanced Fields for R The First Reduction - to a Euclidean R Electric and Magnectic Fields Radially Symmetrie 49

3 X Table of Contents The Use of Computer Graphics for Solving Problems in Singularity Theory Richard J. Morris Introduction A Toolkit for Exploring Singularity Theory Two Problems in Singularity Theory 60 What Should a Surface in 4-Space Look Like? Dennis Roseman Introduction Mathematical Setting What Mathematical Questions Do We Want to Address? How to Improve a Knotted Surface? Visualization in 4-Dimensions: General A Concrete Example Visualization Solutions in M 4 : Projections, Slices, and Slabs Visualization Solutions in R 4 : Adding the "Missing" Information Some Current Visualization Tools Notes on the Figures 79 Animation of Algebraic Surfaces Dietmar Saupe and Matthias Ruhl Introduction Raytracing Algebraic Surfaces Polygonalization and Physically-based Interactive Animation Conclusion 91 Part II. Geometrie Algorithms and Experiments Using Symnietry Features of the Surface Evolver to Study Foams Ken A. Brakke and John M. Sullivan Introduction The Foams of Kelvin and Weaire-Phelan Triply Periodic Foams in the Evolver Using Two-fold Symmetries of Minimal Surfaces for the Kelvin Foam Other Rotational Symmetries and the Weaire-Phelan Foam Using Edge Energies to Eliminate Faces in Mirror Planes 114

4 Table of Contents XI Constant Mean Curvature Surfaces Derived from Delaunay's and Wente's Examples Karsten Große-Brauckmann and Konrad Polthier Introduction Delaunay Unduloids and Wente Tori Modifying the Wente Torus Moduli Spaces of Embedded Symmetrie Surfaces with Delaunay Ends 125 Visualization of Periodic Tilings Daniel H. Huson Tilings and Patterns A Fundamental Data-Structure for Periodic Tilings Visualization of 2-Dimensional Tilings Software 138 An Algorithm for Discrete Constant Mean Curvature Surfaces Bernd Oberknapp and Konrad Polthier Introduction Prerequisites Discrete Harmonie Maps and Minimal Surfaces in Discrete CMC Surfaces via Conjugation Conjugate Surface Construction and Symmetry Numerical Considerations and Visualization 158 Part III. Visualization Algorithms and Data Structures Efncient Calculation of Subdivision Surfaces for Visualization Markus Köhler and Heinrich Müller Introduction Doo-Sabin Subdivision Surfaces Depth-First Subdivision Tracing Strategies Examples 177 Fast Line Integral Convolution for Arbitrary Surfaces in 3D Henrik Battke, Detlev Stalling, and Hans-Christian Hege Introduction Background LIC on Surfaces Projecting 3D-Fields 192

5 XII Table of Contents 5. Applications and Results 193 Visualization of Parallel Data based on Procedural Access Martin Rumpf and Bernhard Schupp Introduction Review of Domain Decomposition and Related Visualization Approaches A Procedural Interface to Arbitrary Meshes The Parallel Case Improving Efficiency by Filtering Aspects of Time Scheduling Conclusions 206 Part IV. Visualization Environments A new 3D Graphics Library: Concepts, Implementation, and Examples Markus Alefeld, Jörg Haber, and Alexander Heim Concepts Implementation Examples 221 A Generic Approach to Computer Graphics EkkehardBeier Introduction Categories of Extensibility Generic-3D Functionality G vs. ISO CGRM Examples How to start with G Conclusion 236 MRT A Visualization Tool Addressing Problems 'outside' the Classical Rendering Domain Dieter W. Fellner Introduction Architecture of MRT MRT at Work Introducing New Objects 245

6 Table of Contents XIII 5. Experiences Conclusions 248 Oorange: A Virtual Laboratory for Experimental Mathematics Charles Gunn, Armin Ortmann, Ulrich Pinkall, Konrad Polthier, and Uwe Schwarz Introduction Overview of Oorange Components of the Virtual Laboratory Special Tools Previous Work Future Directions Conclusion 263 Linear Inductive Reductive Dataflow System for ViSC Jacques Lemordant Introduction Field Object Field Scene Object Inductive Reductive Dataflow Graphical Scene Object Example Conclusions 276 See what I mean? Using Graphics Toolkits to Visualise Numerical Data Jeremy Walton and Michael Dewar Introduction Graphics Toolkits for Visualisation A Computer Algebra Package A Visualisation Web Server Benchmarking Visualisation Systems Conclusions 296 Part V. Visualization and Simulation Techniques Numerical Algorithms and Visualization in Medical Treatment Planning Rudolf Beck, Peter Deuflhard, Hans-Christian Hege, Martin Seebaß, and Detlev Stalling Introduction 303

7 XIV Table of Contents 2. Computer Based Therapy Planning The Medical Planning System HyperPlan Conclusions 324 Level Set Methods for Curvature Flow, Image Enchancement, and Shape Recovery in Medical Images Ravi Malladi and James A. Sethian Introduction Level Set Methods Motion under Curvature, Self-Similar Flows, and Minimal Surfaces Image Enchancement and Noise Removal Shape Recovery 341 Numerical Methods, Simulations and Visualization for Compressible Flows Monika Wierse, Thomas Geßner, and Dietmar Kröner Flow Problem in a Simplified Two-Stroke Engine Numerical Method Graphical Demands Further Numerical Test Problems Analysing the Exchange Process 354 Appendix: Color Plates 357