Anatoly T. Fomenko and Tosiyasu L. Kunii, “Topological Modeling for Visualization“

Preface

Flooding of information on the Earth through various computer networks such as the Internet characterizes the world situation we live. Information worlds often called virtual spaces and cyber spaces have been formed on computer networks. The complexity of information worlds has been increasing almost exponentially through the exponential growth of computer networks. Such non-linearity in growth and in span characterizes information worlds. In other words, the characterization of nonlinearity is the key to understand, utilize and live through the flooding of information. The characterization approach is by characteristic points such as peaks, pits and passes according to the Morse theory. Another approach is by singularity signs such as folds and cusps. Atoms and molecules are the other fundamental characterization approach. Topology and geometry including differential topology serve as the framework of the characterization. Topological Modeling for Visualization is a textbook for people interested in the characterization to understand how to do it and what it is. Understanding is the key to utilize information worlds and to live through the changes of the real world on the Earth.

Writing this textbook paused the authors careful preparation. There are heavy mathematical stuff that requires the design of writing style for easy un- derstanding and for strong attraction. To realize the style, we set the main goal of this book to establish a link between the theoretical aspects of modern geometry and topology, on the one hand, and experimental computer geometry, on the other. There are many excellent books on modern geometry and topology (roughly speaking, “theory”), and many excellent books on modern computer and experimental geometry. But as far as we know, there is no book that bridges the gap between these two branches of modern science, that is, between “theory” and “practice”. We have tried to fill this gap. Our intention was to write a book that will be useful to both communities of scientists. Of course, we realize that this separation between “theoretical science” and “experimental science” is not clear-cut, and we use this language and images only for faster description of our main idea. We collect in the book some basic elements of theoretical geometry and topology that are used today in different branches of experimental computer geometry. We do not give detailed proofs because of lack of space, but we give references that can help the reader find the proofs. The advantage of such a style is this: We collect in one book a short description of the most powerful theoretical tools, and experts in experimental science can use this material in their concrete work. Certainly, as we know from our own experience, modern topological methods can improve the results of experimental computer geometry. On the other hand, experts in theoretical geometry and topology can find in our book possible applications of those fields to very interesting computer experiments in the world of geometrical computer methods, medicine, cars industry, architecture, and so on. Many pure mathematicians will also find here material for development of a new theoretical ideas. Each chapter consists of two layers: first theoretical ideas, then applications to the different branches of modern experimental computer geometry. We’ve tried to make chapters as independent as possible, to help to the reader use each chapter as an individual research tool, without a complete study of other sections of the book. As a consequence, sometimes we repeat in some chapters a summary of material from another section, to recall important notions.