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E-raamat: Classical and Discrete Differential Geometry: Theory, Applications and Algorithms [Taylor & Francis e-raamat]

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  • Formaat: 568 pages, 178 Line drawings, black and white; 112 Halftones, black and white; 290 Illustrations, black and white
  • Ilmumisaeg: 31-Jan-2023
  • Kirjastus: CRC Press
  • ISBN-13: 9781003350576
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  • Taylor & Francis e-raamat
  • Hind: 156,95 €*
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Classical and Discrete Differential Geometry: Theory, Applications and AlgorithmsSuurem pilt
  • Formaat: 568 pages, 178 Line drawings, black and white; 112 Halftones, black and white; 290 Illustrations, black and white
  • Ilmumisaeg: 31-Jan-2023
  • Kirjastus: CRC Press
  • ISBN-13: 9781003350576
Teised raamatud teemal:
Taylor & Francis e-raamatudRohkem infot Taylor & Francis e-raamatute kohta
Raamatu kodulehekülg: https://www.taylorfrancis.com/books/9781003350576
"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 andnetworks. 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, andto 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"--

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.

Preface xv
Section I Differential Geometry, Classical and Discrete
Chapter 1 Curves
3(46)
1.1 Curves
4(2)
1.2 Curvature
6(21)
1.2.1 The Osculating Circle
7(6)
1.2.2 Menger Curvature
13(4)
1.2.2.1 Applications Of Menger Curvature
17(3)
1.2.3 Haantjes Curvature
20(4)
1.2.4 Applications Of Haantjes Curvature
24(3)
1.3 Torsion
27(20)
1.3.1 The Serret-Frenet Formulas
29(6)
1.3.2 Haantjes Curvature Revisited
35(1)
1.3.3 The Local Canonical Form
36(3)
1.3.4 Existence And Uniqueness Theorem
39(1)
1.3.5 Metric Torsion
40(6)
1.3.5.1 The Metric Existence And Uniqueness Theorem Of Curves
46(1)
1.4 Higher Dimensional Curves
47(2)
Chapter 2 Surfaces: Gauss Curvature - First Definition
49(42)
2.1 Surfaces
49(5)
2.2 Gauss Curvature - First Definition
54(9)
2.3 The Fundamental Forms
63(22)
2.3.1 The First Fundamental Form
63(3)
2.3.1.1 Examples
66(2)
2.3.1.2 The Second Fundamental Form
68(8)
2.3.1.3 Distinguished Curves Revisited
76(9)
2.4 Some Implementation Aspects
85(6)
Chapter 3 Metrization Of Gauss Curvature
91(22)
3.1 Metric Approximation Of Sectional Curvatures
91(4)
3.2 Wald Curvature
95(18)
3.2.1 Computation Of Wald Curvature I: The Exact Formula
102(2)
3.2.2 Computation Of Wald Curvature II: An Approximation
104(4)
3.2.3 Applications Of Wald Curvature
108(1)
3.2.4 Wald Curvature Revisited
109(4)
Chapter 4 Gauss Curvature And Theorema Egregium
113(36)
4.1 Theorema Egregium
113(28)
4.1.1 The Tube Formula And Approximation Of Surface Curvatures
131(10)
4.2 Normal Cycle
141(8)
Chapter 5 The Mean And Gauss Curvature Flows
149(8)
5.1 Curve Shortening Flow
149(4)
5.2 Mean Curvature Flow
153(1)
5.3 Gauss Curvature Flow
154(3)
Chapter 6 Geodesies
157(36)
6.1 Covariant Derivative
157(5)
6.2 Geodesics
162(20)
6.2.1 The Hopf-Rinow Theorem
181(1)
6.3 Discretization Of Geodesics
182(11)
Chapter 7 Geodesies And Curvature
193(14)
7.1 Gauss Curvature And Parallel Transport
203(4)
Chapter 8 The Equations Of Compatibility
207(6)
8.1 Applications And Discretizations
211(2)
Chapter 9 The Gauss-Bonnet Theorem And The Poincare Index Theorem
213(26)
9.1 The Gauss-Bonnet Theorem
213(12)
9.1.1 The Local Gauss-Bonnet Theorem
214(2)
9.1.2 The Global Gauss-Bonnet Theorem
216(9)
9.2 The Poincare Index Theorem
225(14)
9.2.1 Discretizations Of The Gauss-Bonnet Theorem
231(4)
9.2.2 Discretizations Of The Poincare Index Theorem
235(4)
Chapter 10 Higher Dimensional Curvatures
239(10)
10.1 Motivation And Basics
239(10)
10.1.1 The Curvature Tensor
240(1)
10.1.2 Sectional Curvature
240(2)
10.1.3 Ricci Curvature
242(5)
10.1.4 Scalar Curvature
247(2)
Chapter 11 Higher Dimensional Curvatures 2
249(18)
11.1 Motivation
249(1)
11.2 The Lipschitz-Killing Curvatures
249(6)
11.2.1 Curvatures' Approximation
251(1)
11.2.1.1 Thick Triangulations
251(2)
11.2.1.2 Curvatures' Approximation Results
253(2)
11.3 Generalized Principal Curvatures
255(1)
11.4 Other Approaches
256(11)
11.4.1 Banchoff's Definition Revisited
256(1)
11.4.2 Stone's Sectional Curvature
257(1)
11.4.3 Glickenstein's Sectional, Ricci And Scalar Curvatures
258(2)
11.4.4 The Ricci Tensor Of Alsing And Miller
260(1)
11.4.5 The Metric Approach
260(1)
11.4.5.1 Metrization Of The Lipschitz-Killing Curvatures
261(2)
11.4.5.2 A Metric Gauss-Bonnet Theorem And Pl Curvatures
263(4)
Chapter 12 Discrete Ricci Curvature And Flow
267(50)
12.1 Pl Manifolds - From Combinatorial To Metric Ricci Curvature
267(15)
12.1.1 Definition And Convergence
268(5)
12.1.2 The Bonnet-Myers Theorem
273(1)
12.1.2.1 The 2-Dimensional Case
273(6)
12.1.2.2 Wald Curvature And Alexandrov Spaces
279(2)
12.1.3 A Comparison Theorem
281(1)
12.2 Ricci Curvature And Flow For 2-Dimensional Pl Surfaces
282(26)
12.2.1 Combinatorial Surface Ricci Flow
282(17)
12.2.2 The Metric Ricci Flow For Surfaces
299(1)
12.2.2.1 Smoothings And Metric Curvatures
300(1)
12.2.2.2 A Metric Ricci Flow
301(5)
12.2.3 Combinatorial Yamabe Flow
306(2)
12.3 Ricci Curvature And Flow For Networks
308(9)
12.3.1 Metric Ricci Curvature Of Networks
308(2)
12.3.2 Ollivier Ricci Curvature
310(7)
Chapter 13 Weighted Manifolds And Ricci Curvature Revisited
317(22)
13.1 Weighted Manifolds
318(6)
13.1.1 The Curvature-Dimension Condition Of Lott-Villani And Sturm
319(2)
13.1.2 Corwin et al.
321(1)
13.1.2.1 Curvature Of Curves In Weighted Surfaces
321(1)
13.1.2.2 The Mean Curvature Of Weighted Surfaces
322(1)
13.1.2.3 Gauss Curvature Of Weighted Surfaces
323(1)
13.2 Forman-Ricci Curvature
324(15)
13.2.1 The General Case
324(2)
13.2.2 Two-Dimensional Complexes
326(7)
13.2.3 The Forman-Ricci Curvature Of Networks
333(3)
13.2.3.1 From Networks To Simplicial Complexes
336(3)
Section II Differential Geometry, Computational Aspects
Chapter 14 Algebraic Topology
339(30)
14.1 Introduction
339(2)
14.2 Surface Topology
341(4)
14.3 Fundamental Group
345(3)
14.4 Word Group Representation
348(1)
14.5 Fundamental Group Canonical Representation
349(7)
14.6 Covering Space
356(4)
14.7 Computational Algorithms
360(9)
Chapter 15 Homology And Cohomology Group
369(24)
15.1 Simplicial Homology
369(6)
15.2 Homology Vs. Homotopy
375(3)
15.3 Simplicial Cohomology
378(4)
15.4 Simplicial Mapping
382(3)
15.5 Fixed Point
385(4)
15.6 Computational Algorithms
389(4)
Chapter 16 Exterior Calculus And Hodge Decomposition
393(28)
16.1 Exterior Differentials
393(4)
16.2 De Rham Cohomology
397(1)
16.3 Hodge Star Operator
398(5)
16.4 Hodge Decomposition
403(5)
16.5 Discrete Hodge Theory
408(13)
Chapter 17 Harmonic Map
421(16)
17.1 Planar Harmonic Maps
421(4)
17.2 Surface Harmonic Maps
425(3)
17.3 Discrete Harmonic Map
428(3)
17.4 Computational Algorithm
431(6)
Chapter 18 Riemann Surface
437(12)
18.1 Riemann Surface
437(3)
18.2 Meromorphic Differential
440(4)
18.3 Riemann-Roch Theorem
444(2)
18.4 Abel-Jacobian Theorem
446(3)
Chapter 19 Conformal Mapping
449(28)
19.1 Topological Quadrilateral
449(4)
19.2 Topological Annulus
453(4)
19.3 Riemann Mapping For Topological Disk
457(2)
19.4 Topological Poly-Annulus Slit Map
459(6)
19.5 Koebe's Iteration For Poly Annulus
465(10)
19.6 Topological Torus
475(2)
Chapter 20 Discrete Surface Curvature Flows
477(20)
20.1 Yamabe Equation
477(2)
20.2 Surface Ricci Flow
479(2)
20.3 Discrete Surface
481(2)
20.4 Discrete Surface Yamabe Flow
483(3)
20.5 Topological Quadrilateral
486(4)
20.6 Topological Annulus
490(2)
20.7 Topological Poly-Annulus
492(2)
20.8 Topological Torus
494(3)
Chapter 21 Mesh Generation Based On Abel-Jacobi Theorem
497(24)
21.1 Quad-Meshes And Meromorphic Quartic Forms
497(7)
21.2 Metrics With Special Holonomies
504(5)
21.3 Mesh Generation
509(12)
Section III Appendices
Appendix A Alexandrov Curvature
521(1)
A.1 Alexandrov Curvature
521(1)
A.2 Alexandrov Curvature Vs. Wald Curvature
522(2)
A.3 Rinow Curvature
524(3)
Appendix B Thick Triangulations Revisited
527(4)
Appendix C The Gromov-Hausdorff Distance
531(8)
Bibliography 539(24)
Index 563
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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