The use of computer graphics for solving problems in singularity theory

Richard Morris (rich@singsurf.org)
University of Liverpool


This paper was presented at the Visulisation of Mathematics Workshop in Berlin (VizMath '95). And appears in "Visualization and Mathematics: Experiments, Simulations and Environments", Hans-Christian Hege, Konrad Polthier (eds), Springer Verlag Heidelberg . ISBN 3-540-61269-6. Full Volume

PDF version of the full paper

Abstract

We explore two investigation in singularity theory in which mathematical visualisation played and 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.

Introduction

Many problems in differential geometry and singularity theory naturally lend themselves to graphical solutions. For example there is considerable interest in calculating ridges and sub-parabolic lines various surfaces which can be readily visualised. In other more theoretical problems, such as classifying the singularities of maps, graphical solutions can play an important part in the solution. Some of the applications in which visualisation have already helped include: symmetry sets and rotational symmetry sets of plane curves \cite{TG_SS,Thesis}; ridges and sub-parabolic lines of surfaces \cite{BGTridge,Thesis,pud}; maps from $\Re^2$ to $\Re^2$ \cite{Gibson}; binary differential equations \cite{BT_binary}; duals of surfaces \cite{BGT_dual,West}; robotic grasp \cite{desmond}; cusp tracking \cite{gordon}; and quartic curves \cite{wall}.

In section~\ref{tool_sec} we will look at a set of programs we have developed to tackle the basic graphical problems which occur in singularity theory. We will discuss some of the user interface issues as well as the algorithms used to create accurate representations of singular surfaces.

In section~\ref{example_sec} we will look at the role experimental results have to play in the development of mathematical proof. We will examine two cases studies which are good examples of this experimental method and illustrate some of the techniques needed for successful experimental analysis. In both cases there was a conflict between mathematical conjectures and the results generated by computer graphics. This conflict caused both the mathematics and the experimental results to be reexamined leading to a resolution of the problems. In the first case a the experimental results were shown to be correct and in the second both methods were partially correct and the final results had a much richer structure.

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