Topological Modeling for Visualization

1. Front Matter

   Pages ii-x

   PDF

2. Foundation

   1. Front Matter

      Pages 1-1

      PDF

   2. Curves

      • Anatolij T. Fomenko, Tosiyasu L. Kinii

      Pages 3-24

   3. The Notion of a Riemannian Metric

      • Anatolij T. Fomenko, Tosiyasu L. Kinii

      Pages 25-33

   4. Local Theory of Surfaces

      • Anatolij T. Fomenko, Tosiyasu L. Kinii

      Pages 35-66

   5. The Classification of Surfaces

      • Anatolij T. Fomenko, Tosiyasu L. Kinii

      Pages 67-94

   6. Abstract Manifolds

      • Anatolij T. Fomenko, Tosiyasu L. Kinii

      Pages 95-104

   7. Critical Points and Morse Theory

      • Anatolij T. Fomenko, Tosiyasu L. Kinii

      Pages 105-125

   8. Application: Analyzing Human Body Motions Using Manifolds and Critical Points

      • Anatolij T. Fomenko, Tosiyasu L. Kinii

      Pages 127-137

   9. Computer Examination of Surfaces and Morse Functions

      • Anatolij T. Fomenko, Tosiyasu L. Kinii

      Pages 139-180

   10. Height Functions and Distance Functions

       • Anatolij T. Fomenko, Tosiyasu L. Kinii

       Pages 181-190

   11. Homotopies and Surface Generation

       • Anatolij T. Fomenko, Tosiyasu L. Kinii

       Pages 191-210

   12. Homology

       • Anatolij T. Fomenko, Tosiyasu L. Kinii

       Pages 211-244

   13. Geodesics

       • Anatolij T. Fomenko, Tosiyasu L. Kinii

       Pages 245-261

   14. Transformation Groups

       • Anatolij T. Fomenko, Tosiyasu L. Kinii

       Pages 263-286

3. Advanced Subjects

   1. Front Matter

      Pages 287-288

      PDF

   2. Computers and Visualization in Hyperbolic Three-Dimensional Geometry and Topology

      • A. T. Fomenko, S. V. Matveev

      Pages 289-312

   3. Integrable Hamiltonian Systems with Two Degrees of Freedom

      • A. V. Bolsinov

      Pages 313-330

   4. Topological and Orbital Analysis of Integrable Lagrange and Goryachev-Chaplygin Problems

      • O. E. Orël

      Pages 331-347

   5. Numerical Calculation of the Orbital Invariant of Goryachev—Chaplygin and Lagrange Systems

      • S. Takahashi

      Pages 349-374

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.

• Riemannian manifold
• calculus
• differential geometry
• geometry
• manifold
• modeling
• topology
• visualization

• Department of Differential Geometry and Applications, Faculty of Mathematics and Mechanics, Moscow State University, Moscow, Russia

  Anatolij T. Fomenko

• Laboratory of Digital Art and Technology, Tokyo, 113, Japan

  Tosiyasu L. Kunii

• Senior Partner of MONOLITH Co. Ltd., Tokyo, 106, Japan

  Tosiyasu L. Kunii

Policies and ethics