Ebook: Topological Modeling for Visualization

The flood of information through various computer networks such as the In­ ternet characterizes the world situation in which we live. Information worlds, often called virtual spaces and cyberspaces, have been formed on computer networks. The complexity of information worlds has been increasing almost exponentially through the exponential growth of computer networks. Such nonlinearity in growth and in scope characterizes information worlds. In other words, the characterization of nonlinearity is the key to understanding, utiliz­ ing and living with the flood 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 ap­ proach. Topology and geometry, including differential topology, serve as the framework for the characterization. Topological Modeling for Visualization is a textbook for those interested in this characterization, to understand what it is and how to do it. Understanding is the key to utilizing information worlds and to living with the changes in the real world. Writing this textbook required careful preparation by the authors. There are complex mathematical concepts that require designing a writing style that facilitates understanding and appeals to the reader. To evolve a style, we set as a main goal of this book the establishment of a link between the theoretical aspects of modern geometry and topology, on the one hand, and experimental computer geometry, on the other.

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Content:
Front Matter....Pages ii-x
Front Matter....Pages 1-1
Curves....Pages 3-24
The Notion of a Riemannian Metric....Pages 25-33
Local Theory of Surfaces....Pages 35-66
The Classification of Surfaces....Pages 67-94
Abstract Manifolds....Pages 95-104
Critical Points and Morse Theory....Pages 105-125
Application: Analyzing Human Body Motions Using Manifolds and Critical Points....Pages 127-137
Computer Examination of Surfaces and Morse Functions....Pages 139-180
Height Functions and Distance Functions....Pages 181-190
Homotopies and Surface Generation....Pages 191-210
Homology....Pages 211-244
Geodesics....Pages 245-261
Transformation Groups....Pages 263-286
Front Matter....Pages 287-288
Computers and Visualization in Hyperbolic Three-Dimensional Geometry and Topology....Pages 289-312
Integrable Hamiltonian Systems with Two Degrees of Freedom....Pages 313-330
Topological and Orbital Analysis of Integrable Lagrange and Goryachev-Chaplygin Problems....Pages 331-347
Numerical Calculation of the Orbital Invariant of Goryachev—Chaplygin and Lagrange Systems....Pages 349-374
Ridges, Ravines and Singularities....Pages 375-383
Back Matter....Pages 385-395

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Content:
Front Matter....Pages ii-x
Front Matter....Pages 1-1
Curves....Pages 3-24
The Notion of a Riemannian Metric....Pages 25-33
Local Theory of Surfaces....Pages 35-66
The Classification of Surfaces....Pages 67-94
Abstract Manifolds....Pages 95-104
Critical Points and Morse Theory....Pages 105-125
Application: Analyzing Human Body Motions Using Manifolds and Critical Points....Pages 127-137
Computer Examination of Surfaces and Morse Functions....Pages 139-180
Height Functions and Distance Functions....Pages 181-190
Homotopies and Surface Generation....Pages 191-210
Homology....Pages 211-244
Geodesics....Pages 245-261
Transformation Groups....Pages 263-286
Front Matter....Pages 287-288
Computers and Visualization in Hyperbolic Three-Dimensional Geometry and Topology....Pages 289-312
Integrable Hamiltonian Systems with Two Degrees of Freedom....Pages 313-330
Topological and Orbital Analysis of Integrable Lagrange and Goryachev-Chaplygin Problems....Pages 331-347
Numerical Calculation of the Orbital Invariant of Goryachev—Chaplygin and Lagrange Systems....Pages 349-374
Ridges, Ravines and Singularities....Pages 375-383
Back Matter....Pages 385-395
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