The main goal of this textbook is to establish a bridge between the theoretical aspects of modern geometry and topology on the one hand and computer experimental geometry on the other. Thus the theory and application of mathematical visualization are given equal emphasis. This, along with the ample illustrations and the fact that each chapter is designed as an independent unit, makes Topological Modeling for Visualization a unique book in its field. The two famous authors, A. T. Fomenko and T. L. Kunii, thoroughly explain the necessary mathematical techniques so that undergraduate students with only a grounding in high-school mathematics can benefit from using this book. While linear problems are covered, the emphasis is on nonlinear problems, with many examples relating to natural phenomena and todays abundant information sources.
Students in the fields of mathematics and computing will find it rigorous enough to serve as a basic text in differential geometry and topology, while students from fields as diverse as cognitive science and economics who need to solve nonlinear problems will find this book indispensable. Contents:
Part I: Foundation: Curves. The Notion of a Riemannian Metric. Local Theory of Surfaces. The Classification of Surfaces. Abstract Manifolds. Critical Points and Morse Theory. Analyzing Human Body Motions Using Manifolds and Critical Points. Computer Examination of Surfaces and Morse Functions. Height Functions and Discrete Functions. Homotopies and Surface Generation. Homology. Geodesics. Transformation Groups.-
Part II: Advanced Subjects: Hyperbolic Geometry and Topology. Hamiltonian System with Two Degrees of Freedom. Topological and Orbital Analysis of Integrable Problems. Orbital Invariant of Integrable Hamiltonian Systems. Ridges, Ravines and Singularities.- Bibliography.- Index.