Skip to main content

Über dieses Buch

Visualization and mathematics have begun a fruitful relationship, establishing links between problems and solutions of both fields. In some areas of mathematics, like differential geometry and numerical mathematics, visualization techniques are applied with great success. However, visualization methods are relying heavily on mathematical concepts.
Applications of visualization in mathematical research and the use of mathematical methods in visualization have been topic of an international workshop in Berlin in June 1995. Selected contributions treat topics of particular interest in current research. Experts are reporting on their latest work, giving an overview on this fascinating new area. The reader will get insight to state-of-the-art techniques for solving visualization problems and mathematical questions.



Visualizing Mathematics


The Minimax Sphere Eversion

We consider an eversion of a sphere driven by a gradient flow for elastic bending energy. We start with a halfway model which is an unstable Willmore sphere with 4-fold orientation-reversing rotational symmetry. The regular homotopy is automatically generated by flowing down the gradient of the energy from the halfway model to a round sphere, using the Surface Evolver. This flow is not yet fully understood; however, our numerical simulations give evidence that the resulting eversion is isotopic to one of Morin’s classical sphere eversions. These simulations were presented as real-time interactive animations in the CAVE TM automatic virtual environment at Supercomputing’95, as part of an experiment in distributed, parallel computing and broad-band, asynchronous networking.
George Francis, John M. Sullivan, Rob B. Kusner, Ken A. Brakke, Chris Hartman, Glenn Chappell

Exploring Plane Hyperbolic Geometry

Hyperbolic geometry is a geometry whose Euclidean representations cannot be conveniently handled. Straight edge and compass are not the best tools for exploring hyperbolic geometry. Interactive software, as described in this paper, is much more appropriate. A good way of finding out about a new mathematical structure is on one hand, to visualize the mathematical objects involved and on the other, to observe how structure preserving mappings work on these objects. Both of these are supported by our software.
Barbara Hausmann, Britta Slopianka, Hans-Peter Seidel

Visualizing Nonlinear Electrodynamics

Visualizing phenomena in high dimensions requires a combination of equivariant mathematics and computer graphics. This paper applies the method of equivariant geometry, which involves Lie Groups and PDE’s, to the study of nonlinear electrodynamics. For the difficult calculations and for the graphics we used Maple.
Geoffrey Martin, Ivan Sterling

The Use of Computer Graphics for Solving Problems in Singularity Theory

We explore two investigations in singularity theory in which mathematical visualisation played an important part in the proof. We also describe a computer package which has been used to aid the experimental investigation of singularity theory and outline some of the computational problems involved in rendering singular surfaces.
Richard J. Morris

What Should a Surface in 4-Space Look Like?

We wish to investigate spaces of dimension greater than three and in particular surfaces in 4-dimensional space. Such surfaces can be knotted. Our explorations include mathematical and visualization tools.
Mathematically we focus on a particular example of visualization of the result of an energy flow of a knotted surface. In terms of visualization, we use sound, texture, and a slicing technique of “splayed slabs”, in addition to more traditional tools.
Basic issues of the mathematical visualization process are discussed.
Dennis Roseman

Animation of Algebraic Surfaces

This contribution is a practise-and-experience report on the visualization of parameter dependent algebraic surfaces. An important example of a deformation of algebraic surfaces was proposed by Kummer in the last century. The techniques in computer graphics algorithms, software and hardware for the animated real-time display of many types of surfaces have been established in recent years. We present an application of this technology and a raytracer to the Kummer family and others. The purpose of this report is to demonstrate the feasibility of such studies by combining several pieces of readily available software and off-the-shelf hardware with only a minimal investment of extra programming. Raytracing produces high quality renderings, however, the method requires much processing time. Faster alternatives are methods yielding polygonalizations or using physically-based approaches. These provide less quality but allow near real-time user interaction. We present some examples for them.
Dietmar Saupe, Matthias Ruhl

Geometric Algorithms and Experiments


Using Symmetry Features of the Surface Evolver to Study Foams

This paper describes the use of various symmetry features, including periodic boundary conditions, mirror boundaries, and rotational symmetry, in the Evolver. As a test case, we use these features to study foams, in particular the equal-volume foams of Kelvin and Weaire-Phelan. To compute the shape and energy of one of these compound surfaces, it is most efficient to work with only the smallest possible fundamental domain.
Ken A. Brakke, John M. Sullivan

Constant Mean Curvature Surfaces Derived from Delaunay’s and Wente’s Examples

We show how Wente tori and Delaunay surfaces can be used as building blocks to construct new surfaces of constant mean curvature. In a first part we give examples of periodic Wente tori and Wente tori with Delaunay ends. In a second part we study all embedded Delaunay-like surfaces with a fixed number of ends and some given reflectional symmetry.
Karsten Große-Brauckmann, Konrad Polthier

Visualization of Periodic Tilings

Delaney symbols provide a fundamental data-structure for periodic tilings. Based on this observation, algorithms have been developed and implemented for the systematic enumeration, visualization and interactive manipulation of periodic tilings of the plane, sphere and hyperbolic plane.
Daniel H. Huson

An Algorithm for Discrete Constant Mean Curvature Surfaces

We present a new algorithm for computing discrete constant mean curvature surfaces in ℝ3. It is based on the definition of a discrete version of the conjugate surface construction for cmc surfaces. Here we solve a Plateau problem for a discrete minimal surface in S3 by computing a sequence of discrete harmonic maps F i : S3 → S3. The definition of a discrete conjugation allows to transform this sequence to a sequence of conjugate discrete maps which converges to a discrete cmc surface in ℝ3. The algorithm is applicable to free boundary value problems for cmc surfaces and led to the recent discovery of new compact cmc surfaces.
Bernd Oberknapp, Konrad Polthier

Visualization Algorithms and Data Structures


Efficient Calculation of Subdivision Surfaces for Visualization

A subdivision surface is defined by a polygonal mesh which is iteratively refined into an infinite sequence of meshes converging to the desired smooth surface. Classical subdivision schemes are those described and analysed by Catmull—Clark and Doo—Sabin. A graphical representation can be obtained by stopping the iteration on a level of refinement sufficient to yield a smooth representation when drawing the mesh on that level. However, the storage requirements of the finest mesh and those on the previous levels can be considerable, that is exponential in the number of iterations, since the number of mesh elements grows by a constant factor from level to level. We overcome this problem by deviating from level-wise breadth-first subdivision by subdividing the mesh locally in a depth-first manner over all levels of iteration. This results in a front of subdivision which moves over the surface and successively reports the elements of the finest mesh. Only the front of subdivision must be held in main memory, and it needs only about square-root of the space required by the standard method, at about the same time of computation.
Markus Kohler, Heinrich Müller

Fast Line Integral Convolution for Arbitrary Surfaces in 3D

We describe an extension of the line integral convolution method (LIC) for imaging of vector fields on arbitrary surfaces in 3D space. Previous approaches were limited to curvilinear surfaces, i.e. surfaces which can be parametrized globally using 2D-coordinates. By contrast our method also handles the case of general, possibly multiply connected surfaces. The method works by tesselating a given surface with triangles. For each triangle local euclidean coordinates are defined and a local LIC texture is computed. No scaling or distortion is involved when mapping the texture onto the surface. The characteristic length of the texture remains constant.
In order to exploit the texture hardware of modern graphics computers we have developed a tiling strategy for arranging a large number of triangular texture pieces within a single rectangular texture image. In this way texture memory is utilized optimally and even large textured surfaces can be explored interactively.
Henrik Battke, Detlev Stalling, Hans-Christian Hege

Visualization of Parallel Data based on Procedural Access

Large scale applications in numerical continuum mechanics and especially in computational fluid dynamics require enormous computing power and extensive storage. For post processing purposes and to support visual debugging a flexible toolbox of visualization methods addressing the user data directly in the distributed environment is essential. A concept based on procedural access to the data on arbitrary meshes is presented. It leads to an inheritance of a large group of visualization methods developed for scalar data, and is well suited for debugging and visualization during code and data structure construction. Changing the computational mesh representation the user has to adapt his specific procedural interface. Thereby he does not need to know about the visualization routines. Similarly, solely the communication interface has to be changed if a new parallel machine is added to the configuration. But this is decoupled from the visualization itself and the grid type. The presented approach can efficiently be implemented. Aspects and results from an implementation under PVM [13] are discussed.
Martin Rumpf, Bernhard Schupp

Visualization Environments


A new 3D Graphics Library: Concepts, Implementation, and Examples

3D graphics libraries play an important role in aiding both scientists and engineers to visualize their data and results. Existing graphics libraries provide a wide range of applications. However, they lack a common interface and more flexibility in the details.
This article presents a new 3D graphics library, which combines both the speed of hardware driven rendering and the image quality of professional commercial products. This improvement was achieved by a flexible and extendible concept which integrates the use of different renderer algorithms, user-definable shading algorithms, and adaptations to many different hardware platforms.
Based on a catalog of requirements, a modular object-oriented design for a new 3D graphics library has been developed. Some aspects of the resulting implementation are discussed in detail. Several examples of applications built with the help of the new library conclude this article.
Markus Alefeld, Jörg Haber, Alexander Heim

A Generic Approach to Computer Graphics

This paper describes a generic approach to computer graphics. After many experiences with various graphics systems I found that there is a need for a generic graphics kernel consisting of well-defined, well-separated minimal interfaces, which may be used to implement a customized graphics kernel by adding definitions and specific functionality.
The way proposed allows the definition of arbitrary graphics kernels highly optimized for a given task and leaves behind general-purpose systems as commonly used today. The intersection between these kernels could be a well-defined, well-documented generic graphics kernel called Generic-3D.
Ekkehard Beier

MRT — A Visualization Tool Addressing Problems ‘outside’ the Classical Rendering Domain

We present an object-oriented software architecture for a 3D rendering environment which drastically improves the programmer’s productivity, and, most importantly, consists of building blocks that lend themselves to customization thus making 3D image synthesis more accessible.
The rendering platform MRT is object-based rather than drawing based and consists of an extensible set of objects that perform a variety of operations. Experiences with our (inhomogeneous) user population prove that the system meets its design goal of being highly customizable and extendable. Furthermore, it serves as a compact testbed for various rendering aspects as well as for new algorithms ‘outside’ of the classical rendering domain.
This is supported by a recent cooperation with a German mobile communication network supplier. The development of a prototype package to simulate the 3D distribution of radio waves in urban environments based on MRT could be completed by one of our students within three weeks. The incredibly short development time (considering that we started from scratch) in combination with the fact that the prototype was significantly faster than what was available before was a convincing argument to start this cooperation.
Dieter W. Fellner

Oorange: A Virtual Laboratory for Experimental Mathematics

Oorange is a virtual laboratory for experimental mathematics. It consists of a set of infrastructure services supporting the creation, execution, and dissemination of mathematical experiments. For each component of a traditional physical experiment, there is a corresponding Oorange infrastructure feature:
  • Object of study: High level software classes
  • Laboratory equipment: Foundation software classes and function libraries
  • Configuration of specific experiment: Computational network composed of objects
  • Monitor and control: Object inspection; 2D and 3D viewers
  • Running the experiment: Animation objects
  • Recording the experiment: Archiving and scripting
  • Disseminating result: Documentation
A hybrid language scheme underlies the design: interpreted scripts in Tcl manage tasks requiring high flexibility, while a compiled object library in Objective C supports the underlying mathematical algorithms. The resulting system is intended to be accessible to wide range of expertise levels. Oorange is free software distributed according to a GNU-like license agreement.
Charles Gunn, Armin Ortmann, Ulrich Pinkall, Konrad Polthier, Uwe Schwarz

Linear Inductive Reductive Dataflow System for ViSC

We describe a framework for Visualization in Scientific Computing (ViSC). Our interest is in visualizing time dependent multidimensional data. Our proposal is to isolate the data reduction process from the graphical presentation process and to consider time as a continuous variable, treating it as a first class object. The framework is based on the following three models:
  • A scientific data model, derived from the theory of fiber bundles, encapsulating multidimensional data over time and implemented as a cluster of classes.
  • A graphical data model which is a set of time dependent objects encapsulating shape and shading. This model appears to the external world as a set of independent variables.
  • A linear inductive reductive dataflow model that supports the progressive reduction of scientific variables and the progressive computation of graphical variables.
Jacques Lemordant

See what I mean? Using Graphics Toolkits to Visualise Numerical Data

The use of graphics toolkits to visualise and understand numerical data is explored with reference to (a) a common data format for 3D geometry and (b) the improvement of visualisation algorithms through the incorporation of numerical library software.
Using a common 3D format allows scenes and objects to be shared between applications and—if desired—published on the World-Wide Web (WWW) for viewing by co-workers elsewhere. Two examples of this are given. In one, we outline how the adoption of a standard toolkit to provide the visualisation component of a computer algebra package has cut down on development time and has provided it with the ability to share 3D data with other applications. In the other, we show how a graphics toolkit can be used within a visualisation web server, where its output can be transmitted across the WWW by means of the 3D format.
The use of library software can save the application developer time and effort in implementing fundamental algorithms, and allows them to concentrate on other aspects of the visualisation process. We describe some preliminary work on visualisation benchmarking, and show how some library routines are superior to simpler, but less sophisticated algorithms in the context of particle tracing.
Jeremy Walton, Michael Dewar

Visualization and Simulation Techniques


Numerical Algorithms and Visualization in Medical Treatment Planning

After a short summary on therapy planning and the underlying technologies we discuss quantitative medicine by giving a short overview on medical image data, summarizing some applications of computer based treatment planning, and outlining requirements on medical planning systems. Then we continue with a description of our medical planning system HyperPlan. It supports typical working steps in therapy planning, like data aquisition, segmentation, grid generation, numerical simulation and optimization, accompanying these with powerful visualization and interaction techniques.
Rudolf Beck, Peter Deuflhard, Hans-Christian Hege, Martin Seebaß, Detlev Stalling

Level Set Methods for Curvature Flow, Image Enchancement, and Shape Recovery in Medical Images

Level set methods are powerful numerical techniques for tracking the evolution of interfaces moving under a variety of complex motions. They are based on computing viscosity solutions to the appropriate equations of motion, using techniques borrowed from hyperbolic conservation laws. In this paper, we review some of the applications of this work to curvature motion, the construction of minimal surfaces, image enhancement, and shape recovery. We introduce new schemes for denoising three-dimensional shapes and images, as well as a fast shape recovery techniques for three-dimensional images.
Ravi Malladi, James A. Sethian

Numerical Methods, Simulations and Visualization for Compressible Flows

In this paper we want to describe how we solved an industrial flow problem, the simulation in a simplified two—stroke engine. The numerical method we are using is an upwind finite volume method on an unstructured grid with arbitrary finite volumes. The grid fineness is adapted locally to a get a better resolution of important flow structures. The boundaries of the calculation domain are moving. Essential for the development for such a numerical algorithm is the graphical support to find programming errors and of course to visualize the calculated data. We will analyse the demands for the necessary graphics, which is provided by the graphical programming environment GRAPE. Furthermore we will present some of the numerical results of standard test problems in 2D and 3D to validate our numerical algorithm.
Monika Wierse, Thomas Geßner, Dietmar Kröner


Weitere Informationen