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Classical and Discrete Differential Geometry: Theory, Applications and Algorithms book cover

1st Edition

Classical and Discrete Differential Geometry Theory, Applications and Algorithms

By David Xianfeng Gu, Emil Saucan Copyright 2023
    588 Pages 290 B/W Illustrations
    by CRC Press

    588 Pages 290 B/W Illustrations
    by CRC Press
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    This book introduces differential geometry and cutting-edge findings from the discipline by incorporating both classical approaches and modern discrete differential geometry across all facets and applications, including graphics and imaging, physics and networks.

    With curvature as the centerpiece, the authors present the development of differential geometry, from curves to surfaces, thence to higher dimensional manifolds; and from smooth structures to metric spaces, weighted manifolds and complexes, and to images, meshes and networks. The first part of the book is a differential geometric study of curves and surfaces in the Euclidean space, enhanced while the second part deals with higher dimensional manifolds centering on curvature by exploring the various ways of extending it to higher dimensional objects and more general structures and how to return to lower dimensional constructs. The third part focuses on computational algorithms in algebraic topology and conformal geometry, applicable for surface parameterization, shape registration and structured mesh generation.

    The volume will be a useful reference for students of mathematics and computer science, as well as researchers and engineering professionals who are interested in graphics and imaging, complex networks, differential geometry and curvature.

    Section I Differential Geometry, Classical and Discrete  1. Curves  2. Surfaces: Gauss Curvature – First Definition  3. Metrization of Gauss Curvature  4. Gauss Curvature and Theorema Egregium  5. The Mean and Gauss Curvature Flows  6. Geodesics  7. Geodesics and Curvature  8. The Equations of Compatibility  9. The Gauss-Bonnet Theorem and the Poincare Index Theorem  10. Higher Dimensional Curvatures  11. Higher Dimensional Curvatures  12. Discrete Ricci Curvature and Flow  13. Weighted Manifolds and Ricci Curvature Revisited  Section II Differential Geometry, Computational Aspects  14. Algebraic Topology  15. Homology and Cohomology Group  16. Exterior Calculus and Hodge Decomposition 17. Harmonic Map  18. Riemann Surface  19. Conformal Mapping  20. Discrete Surface Curvature Flows  21. Mesh Generation Based on Abel-Jacobi Theorem  Section III Appendices  22. Appendix A  23. Appendix B  24. Appendix C

    Biography

    David Xianfeng Gu is a SUNY Empire Innovation Professor of Computer Science and Applied Mathematics at State University of New York at Stony Brook, USA. His research interests focus on generalizing modern geometry theories to discrete settings and applying them in engineering and medical fields and recently on geometric views of optimal transportation theory. He is one of the major founders of an interdisciplinary field, Computational Conformal Geometry.

    Emil Saucan is Associate Professor of Applied Mathematics at Braude College of Engineering, Israel. His main research interest is geometry in general (including Geometric Topology), especially Discrete and Metric Differential Geometry and their applications to Imaging and Geometric Design, as well as Geometric Modeling. His recent research focuses on various notions of discrete Ricci curvature and their practical applications.

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