计算共形几何(英文原版)

. 2.5 homotopy invariant
2.6 covering spaces
2.7 group representation
2.8 seifert-van kampen theorem
problems
homology and cohomology
3.1 simplicial homology
3.1.1 simplicial complex
3.1.2 geometric approximation accuracy
3.1.3 chain complex
3.1.4 chain map and induced homomorphism
3.1.5 simplicial map
3.1.6 chain homotopy
3.1.7 homotopy equivalence
3.1.8 relation between homology group and homotopy grou
3.1.9 lefschetz fixed point
3.1.10 mayer-vietoris homology sequence
3.1.11 tunnel loop and handle loop
3.2 cohomology
3.2.1 cohomology group
3.2.2 cochain map
3.2.3 cochain homotopy
problems
4 exterior differential calculus
4.1 smooth manifold
4.2 differential forms
4.3 integration
4.4 exterior derivative and stokes theorem
4.5 de rham cohomology group
4.6 harmonic forms
4.7 hodge theorem
problems
5 differential geometry of surfaces
5.1 curve theory
5.2 local theory of surfaces
5.2.1 regular surface
5.2.2 first fundamental form
5.2.3 second fundamental form
5.2.4 weingarten transformation
5.3 orthonormal movable frame
5.3.1 structure equation
5.4 covariant differentiation
5.4.1 geodesic curvature
5.5 gauss-bonnet theorem
5.6 index theorem of tangent vector field
5.7 minimal surface
5.7.1 weierstrass representation
5.7.2 costa minimal surface
problems
riemann surface
6.1 riemann surface
6.2 riemann mapping theorem
6.2.1 conformal module
6.2.2 quasi-conformal mapping
6.2.3 holomorphic mappings
6.3 holomorphic one-forms
6.4 period matrix
6.5 riemann-roch theorem
6.6 abel theorem
6.7 uniformization
6.8 hyperbolic riemann surface
6.9 teichmiiller space
6.9.1 quasi-conformal map
6.9.2 extremal quasi-conformal map
6.10 teichm011er space and modular space
6.10.1 fricke space model
6.10.2 geodesic spectrum
problems
harmonic maps and surface ricci flow
7.1 harmonic maps of surfaces
7.1.1 harmonic energy and harmonic maps
7.1.2 harmonic map equation
7.1.3 rad6's theorem
7.1.4 hopf differential
7.1.5 complex form
7.1.6 bochner formula
7.1.7 existence and regularity
7.1.8 uniqueness
7.2 surface ricci flow
7.2.1 conformal deformation
7.2.2 surface ricci flow
problems
geometric structure
8.1 (x, g) geometric structure
8.2 development and holonomy
8.3 affine structures on surfaces
8.4 spherical structure
8.5 euclidean structure
8.6 hyperbolic structure
8.7 real projective structure
problems
part ii algorithms
topological algorithms
9.1 triangular meshes
9.1.1 half-edge data structure
9.1.2 code samples
9.2 cut graph
9.3 fundamental domain
9.4 basis of homotopy group
9.5 gluing two meshes
9.6 universal covering space
9.7 curve lifting
9.8 homotopy detection
9.9 the shortest loop
9.10 canonical homotopy group generator
further readings
problems
10 algorithms for harmonic maps
10.1 piecewise linear functional space, inner product and laplacian
10.2 newton's method for open surface
10.3 non-linear heat diffusion for closed surfaces
10.4 riemann mapping
10.5 least square method for solving beltrami equation
10.6 general surface mapping
further readings
problems
11 harmonic forms and holomorphic forms
11.1 characteristic forms
11.2 wedge product
11.3 characteristic 1-form
11.4 computing cohomology basis
l 1.5 harmonic 1-form
11.6 hodge star operator
11.7 holomorphic 1-form
11.8 inner product among 1-forms
11.9 holomorphic forms on surfaces with boundaries
11.10 zero points and critical trajectories
11.11 flat metric induced by holomorphic 1-forms
11.12 conformal invariants
11.13 conformal mappings for multi-holed annuli
further readings
problems
12 discrete ricci flow
12.1 circle packing metric
12.2 discrete gaussian curvature
12.3 discrete surface ricci flow
12.4 newton's method
12.5 isometric planar embedding
12.6 surfaces with boundaries
12.7 optimal parameterization using ricci flow
12.8 hyperbolic ricci flow
12.9 hyperbolic embedding
12.9.1 poincare disk model
12.9.2 embedding the fundamental domain
12.9.3 hyperbolic embedding of the universal covering space
12.10 hyperbolic ricci flow for surfaces with boundaries
further readings
problems
a major algorithms
b acknowledgement
reference
index