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E-raamat: Generalized Barycentric Coordinates in Computer Graphics and Computational Mechanics

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  • Formaat: 338 pages
  • Ilmumisaeg: 30-Oct-2017
  • Kirjastus: CRC Press Inc
  • Keel: eng
  • ISBN-13: 9781498763615
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Generalized Barycentric Coordinates in Computer Graphics and Computational MechanicsSuurem pilt
  • Formaat: 338 pages
  • Ilmumisaeg: 30-Oct-2017
  • Kirjastus: CRC Press Inc
  • Keel: eng
  • ISBN-13: 9781498763615
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In Generalized Barycentric Coordinates in Computer Graphics and Computational Mechanics, eminent computer graphics and computational mechanics researchers provide a state-of-the-art overview of generalized barycentric coordinates. Commonly used in cutting-edge applications such as mesh parametrization, image warping, mesh deformation, and finite as well as boundary element methods, the theory of barycentric coordinates is also fundamental for use in animation and in simulating the deformation of solid continua. Generalized Barycentric Coordinates is divided into three sections, with five chapters each, covering the theoretical background, as well as their use in computer graphics and computational mechanics. A vivid 16-page insert helps illustrating the stunning applications of this fascinating research area.

Key Features:











Provides an overview of the many different types of barycentric coordinates and their properties.











Discusses diverse applications of barycentric coordinates in computer graphics and computational mechanics.











The first book-length treatment on this topic
Preface xv
Contributors xix
Section I Theoretical Foundations 1(94)
Chapter 1 Barycentric Coordinates and Their Properties
3(20)
Dmitry Anisimov
1.1 Introduction
4(2)
1.1.1 Barycentric coordinates for simplices
5(1)
1.1.2 Generalized barycentric coordinates
5(1)
1.2 2D Coordinates
6(14)
1.2.1 Wachspress coordinates
7(1)
1.2.2 Discrete harmonic coordinates
8(1)
1.2.3 Mean value coordinates
8(1)
1.2.4 Complete family of coordinates
9(1)
1.2.5 Metric coordinates
9(1)
1.2.6 Poisson coordinates
10(1)
1.2.7 Gordon-Wixom coordinates
10(1)
1.2.8 Harmonic coordinates
11(1)
1.2.9 Maximum entropy coordinates
11(1)
1.2.10 Local coordinates
12(1)
1.2.11 Affine coordinates
12(1)
1.2.12 Sibson coordinates
12(1)
1.2.13 Laplace coordinates
13(1)
1.2.14 Hermite coordinates
13(1)
1.2.15 Complex coordinates
14(1)
1.2.16 Comparison
14(6)
1.3 3D Coordinates
20(4)
1.3.1 Wachspress coordinates
20(1)
1.3.2 Discrete harmonic coordinates
21(1)
1.3.3 Mean value coordinates
21(1)
1.3.4 Complete family of coordinates
22(1)
1.3.5 Other coordinates
22(1)
Chapter 2 Shape Quality for Generalized Barycentric Interpolation
23(20)
Andrew Gillette
Alexander Rand
2.1 Introduction
24(2)
2.2 Deconstructing The A Priori Error Estimate
26(3)
2.3 Shape Quality Metrics For Simplices
29(2)
2.4 Shape Quality Metrics For Polygons And Polyhedra
31(3)
2.5 Interpolation Error Estimates On Polygons
34(5)
2.5.1 Triangulation coordinates
34(1)
2.5.2 Harmonic coordinates
35(2)
2.5.3 Wachspress coordinates
37(2)
2.5.4 Mean value coordinates
39(1)
2.6 Interpolation Error Estimates On Polyhedra And Polytopes
39(2)
2.6.1 Harmonic coordinates in 3D and higher
40(1)
2.6.2 Wachspress coordinates in 3D and higher
40(1)
2.7 Extensions And Future Directions
41(2)
Chapter 3 Transfinite Barycentric Coordinates
43(20)
Alexander G. Belyaev
Pierre-Alain Fayolle
3.1 Introduction
44(1)
3.2 Weighted Mean Value Interpolation
45(6)
3.2.1 General construction
45(2)
3.2.2 Transfinite three-point coordinates
47(1)
3.2.3 Transfinite Laplace coordinates
48(1)
3.2.4 Transfinite Wachspress coordinates
49(2)
3.2.5 Transfinite Laplace and Wachspress coordinates coincide for a disk
51(1)
3.3 Gordon-Wixom Interpolation
51(6)
3.3.1 Lagrange-type Gordon-Wixom interpolation
51(3)
3.3.2 Hermite-type Gordon-Wixom interpolation
54(1)
3.3.3 Modified Hermite-type Gordon-Wixom interpolation
55(1)
3.3.4 Modified Gordon-Wixom for polyharmonic interpolation
56(1)
3.4 Generalized Mean Value Potentials And Distance Function Approximations
57(7)
3.4.1 Generalized mean value potentials for smooth domains
57(4)
3.4.2 Generalized potentials for polygons
61(2)
Chapter 4 Barycentric Mappings
63(14)
Teseo Schneider
4.1 Introduction
64(2)
4.1.1 Convex polygons
64(1)
4.1.2 Arbitrary polygons
65(1)
4.2 Bijective Barycentric Mapping
66(2)
4.2.1 Perturbed target polygons
66(2)
4.3 Bijective Composite Barycentric Mapping
68(3)
4.3.1 Limit of composite barycentric mappings
70(1)
4.4 Extensions
71(4)
4.4.1 Closed planar curves
71(2)
4.4.2 Polyhedra
73(2)
4.5 Practical Considerations
75(2)
4.5.1 Choosing the coordinates
75(1)
4.5.2 Choosing the vertex paths
75(2)
Chapter 5 A Primer on Laplacians
77(18)
Max Wardetzky
5.1 Introduction
77(2)
5.1.1 Basic properties
78(1)
5.2 Laplacians On Riemannian Manifolds
79(4)
5.2.1 Exterior calculus
79(2)
5.2.2 Hodge decomposition
81(1)
5.2.3 The spectrum
82(1)
5.3 Discrete Laplacians
83(14)
5.3.1 Laplacians on graphs
83(1)
5.3.2 The spectrum
84(2)
5.3.3 Laplacians on simplicial manifolds
86(1)
5.3.4 Strongly and weakly defined Laplacians
87(1)
5.3.5 Hodge decomposition
88(1)
5.3.6 The cotan Laplacian and beyond
88(2)
5.3.7 Discrete versus smooth Laplacians
90(5)
Section II Applications in Computer Graphics 95(82)
Chapter 6 Mesh Parameterization
97(12)
Bruno Levy
6.1 Introduction
97(1)
6.2 Applications Of Mesh Parameterization
98(2)
6.3 Notions Of Topology
100(2)
6.4 Tutte's Barycentric Mapping Theorem
102(4)
6.5 Solving The Linear Systems
106(2)
6.6 Choosing The Weights
108(1)
Chapter 7 Planar Shape Deformation
109(26)
Ofir Weber
7.1 Introduction
110(1)
7.2 Complex Barycentric Coordinates
111(12)
7.2.1 Holomorphic functions
113(5)
7.2.2 General construction of complex barycentric coordinates
118(3)
7.2.3 Magic coordinates
121(2)
7.3 Variational Barycentric Coordinates
123(4)
7.3.1 Point-based barycentric maps
123(1)
7.3.2 Point-to-point barycentric coordinates
124(3)
7.4 Conformal Maps
127(5)
7.4.1 Log derivative construction
127(3)
7.4.2 Shape interpolation
130(1)
7.4.3 Variational conformal maps
131(1)
7.5 Implementation Details
132(4)
7.5.1 Visualizing planar maps
132(3)
Chapter 8 Multi-Sided Patches via Barycentric Coordinates
135(12)
Scott Schaefer
8.1 Introduction
136(2)
8.1.1 Bezier form of curves
136(1)
8.1.2 Evaluation
137(1)
8.1.3 Degree elevation
137(1)
8.2 Multisided Bezier Patches In Higher Dimensions
138(5)
8.2.1 Indexing for S-patches
139(2)
8.2.2 Evaluation
141(1)
8.2.3 Degree elevation
142(1)
8.3 Applications
143(4)
8.3.1 Surface patches
143(1)
8.3.2 Spatial deformation
144(3)
Chapter 9 Generalized Triangulations
147(10)
Pooran Memari
9.1 Introduction
147(1)
9.2 Generalized Primal-Dual Triangulations
148(3)
9.2.1 Some classical examples
149(1)
9.2.2 Primal-dual triangulations (PDT)
150(1)
9.3 A Characterization Theorem
151(2)
9.3.1 Combinatorially regular triangulations (CRT)
151(1)
9.3.2 Equivalence between PDT and CRT
152(1)
9.3.3 Parametrization of primal-dual triangulations
152(1)
9.4 Discrete Representation Using Generalized Triangulations
153(5)
9.4.1 Discrete exterior calculus framework
153(1)
9.4.2 Applications in mesh optimization
154(3)
Chapter 10 Self-Supporting Surfaces
157(20)
Etienne Vouga
10.1 Introduction And Historical Overview
158(4)
10.1.1 What is a masonry structure?
158(1)
10.1.2 Heyman's safe theorem
159(1)
10.1.3 Gaudi and hanging nets
160(1)
10.1.4 Maxwell's reciprocal diagrams
161(1)
10.2 Smooth Theory
162(4)
10.2.1 Equilibrium equations
163(1)
10.2.2 Airy stress potential
164(1)
10.2.3 Relative curvatures
164(1)
10.2.4 Curvature interpretation of equilibrium
165(1)
10.3 Discrete Theory
166(4)
10.3.1 Thrust network analysis
166(1)
10.3.2 Thrust networks as block networks
167(1)
10.3.3 Discrete Airy stress potential
167(2)
10.3.4 FEM discretization of Airy stress
169(1)
10.3.5 Discrete curvature interpretation of equilibrium
169(1)
10.4 Optimizing For Stability
170(5)
10.4.1 Alternating optimization
170(2)
10.4.2 Dual formulation as vertex weights
172(1)
10.4.3 Perfect Laplacian optimization
172(1)
10.4.4 Relative-curvature-based smoothing
173(1)
10.4.5 Steel-glass structures and PQ faces
174(1)
10.4.6 Block layouts from stable surfaces
175(1)
10.5 Conclusion And Open Problems
175(5)
10.5.1 Sensitivity analysis of masonry structures
176(1)
10.5.2 Progressive stable structures
176(1)
Section III Applications in Computational Mechanics 177(104)
Chapter 11 Applications of Polyhedral Finite Elements in Solid Mechanics
179(18)
Joseph E. Bishop
11.1 Introduction
180(3)
11.2 Governing Equations Of Solid Mechanics
183(1)
11.3 Polyhedral Finite Element Formulation
184(3)
11.3.1 Weak form of governing equations
185(1)
11.3.2 Shape functions
185(1)
11.3.3 Element integration
186(1)
11.4 Rapid Engineering Analysis
187(4)
11.5 Fragmentation Modeling
191(4)
11.5.1 Fracture methodology
192(1)
11.5.2 Random Voronoi meshes
193(1)
11.5.3 Fragmentation
194(1)
11.6 Summary And Outlook
195(2)
Chapter 12 Extremely Large Deformation with Polygonal and Polyhedral Elements
197(32)
Glaucio H. Paulino
Heno Chi
Cameron Talischi
Oscar Lopez-Pamies
12.1 Introduction
198(2)
12.2 Finite Elasticity Formulations
200(3)
12.2.1 Displacement-based formulation
201(1)
12.2.2 A general two-field mixed variational formulation
201(2)
12.3 Polygonal And Polyhedral Approximations
203(2)
12.3.1 Displacement space on polygons in 2D
203(1)
12.3.2 Displacement space on polyhedra in 3D
204(1)
12.3.3 Pressure space on polygons in 2D
204(1)
12.4 Quadrature Rules And Accuracy Requirements
205(3)
12.5 Gradient Correction Scheme And Its Properties
208(5)
12.5.1 Gradient correction for scalar problems
208(4)
12.5.2 Gradient correction for vectorial problems
212(1)
12.6 Conforming Galerkin Approximations
213(1)
12.7 Numerical Examples
214(7)
12.7.1 Displacement-based polygonal and polyhedral elements
214(3)
12.7.2 Two-field mixed polygonal elements
217(4)
12.8 Application To The Study Of Filled Elastomers
221(9)
12.8.1 Results for filled neo-Hookean elastomers
222(1)
12.8.2 Results for a filled silicone elastomer
223(6)
Chapter 13 Maximum-Entropy Meshfree Coordinates in Computational Mechanics
229(16)
Marino Arroyo
13.1 Introduction
230(1)
13.2 Selecting Barycentric Coordinates Through Entropy Maximization
231(3)
13.3 Introducing Locality: Local Maximum-Entropy Approximants
234(4)
13.4 Further Extensions
238(2)
13.5 Applications
240(3)
13.5.1 High-order partial differential equations
240(1)
13.5.2 Manifold approximation
241(2)
13.6 Outlook
243(2)
Chapter 14 BEM-Based FEM
245(18)
Steffen Weiber
14.1 Introduction
246(1)
14.2 High-Order BEM-Based FEM In 2D
247(6)
14.2.1 Construction of basis functions
248(1)
14.2.2 Finite element method
249(1)
14.2.3 Introduction to boundary element methods
250(2)
14.2.4 Numerical examples
252(1)
14.3 Adaptive BEM-Based FEM In 2D
253(5)
14.3.1 Adaptive FEM strategy
254(1)
14.3.2 Residual-based error estimate for polygonal meshes
255(1)
14.3.3 Numerical examples
256(2)
14.4 Developments And Outlook
258(6)
14.4.1 Hierarchical construction for 3D problems
259(2)
14.4.2 Convection-adapted basis functions in 3D
261(2)
Chapter 15 Virtual Element Methods for Elliptic Problems on Polygonal Meshes
263(18)
Andrea Cangiani
Oliver J. Sutton
Vitaliy Gyrya
Gianmarco Manzini
15.1 Introduction
264(1)
15.2 Virtual Element Spaces And GBC
265(8)
15.2.1 Generalities
265(1)
15.2.2 Lowest order discrete space
265(1)
15.2.3 Generalization to arbitrary order discrete spaces
266(1)
15.2.4 The natural basis
267(2)
15.2.5 A convenient basis
269(2)
15.2.6 A link between the bases
271(2)
15.2.7 Extension to three dimensions
273(1)
15.3 Virtual Element Method For Elliptic PDES
273(3)
15.3.1 Model problem
273(1)
15.3.2 Overview of the conforming VEM
274(2)
15.4 Connection With Other Methods
276(5)
15.4.1 Polygonal and polyhedral finite element method
276(1)
15.4.2 Nodal MFD method
276(2)
15.4.3 BEM-based FEM
278(3)
Bibliography 281(28)
Index 309
Kai Hormann is a full professor in the Faculty of Informatics at USI (Università della Svizzera italiana). His research interests are focused on the mathematical foundations of geometry processing algorithms as well as their applications in computer graphics and related fields. In particular, he is working on generalized barycentric coordinates, subdivision of curves and surfaces, barycentric rational interpolation, and dynamic geometry processing.

N Sukumar is a full professor in the Department of Civil and Environmental Engineering at UC Davis. His research interests are in the areas of computational solid mechanics and applied mathematics, with emphasis on developing and advancing modern finite element and meshfree methods for applications in the deformation and fracture of solids and in ab initio quantum-mechanical materials calculations.
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