Surface Reconstruction Topology for Computing
Index
, 51 ∩
, ∪, 15 ∇
, 88 ≈
, 19 ≃
, 36 ≤
, ≥, 24, 28 ∼
, 18, 74 ⊂
, ⊃, 15 1-1, 15
Abelian, 43 abstract simplicial complex, see simplicial
complex, abstract adjacency theorem, 115
affine combination, 23 Alexander duality, 75
alpha complex, 36 associative, 42
atlas, 21
basis change theorem, 142 basis of neighborhoods, 17
Betti number, 51, 74, 97 β, see Betti number
bijective, 15 binary operation, 42
boundary, 72
group, see group, boundary homomorphism, 71
manifold, 20 set, 16
cancellation, 115 canonical, 121, 128
Cartesian, 14 category, 65
cell complex, 89 chain complex, 72
chain group, see group, chain chart, 19
χ, see Euler characteristic C
∞
, 21 closed set, 16
closure, 16, 29 CMY color model, 157
codomain, 15 coface, 24
collision theorem, 133 column-echelon form, 140
combination, 23 commutative, 42
compact, 20 component tree, 172
composite function, 15 conflict, 153
connected sum, 62 convex combination, 23
convex hull, 23 coordinate function
Cartesian, 14 chart, 20
coset, 46 coset multiplication, 51
covering, 20 critical, 85
cycle, 72 cycle group, see group, cycle
cyclic group, 49
deformation retraction, 35 degenerate, 86
Dehn surgery, 69 derivative, 85
diagonal slide, 166 differential, 85
dimension
chart, 20 manifold, 20
simplicial complex, 24 vector space, 58
240
direct product, 50 distance function, 17
domain, 15 dual complex, 35
echelon form theorem, 141 edge-flip, 166
elementary operations, 137 embedding, 22
epimorphism, 48 equivalence
class, 19 knot, 117
relation, 18 Euclidean metric, 18
Euclidean space, see space, Euclidean Euler characteristic
chain complex, 77 simplicial complex, 61
Euler-Poincaré, 77 face, 24
factor group, 51 field, 56
filtered complex, 32 filtration, 32
alpha complex, 37 manifold sweep, 40
finite type, 102 finitely generated, 50
forking, 109 free Abelian group, 54
function, 15–16 functor, 65
fundamental group, see group, fundamental fundamental theorem of finitely generated
Abelian groups, 50 generator, 49
genus, 64 geometric realization, 27
graded ring and module, 58 gradient, 88
group, 43
boundary, 72 chain, 71
cycle, 72 Fundamental, 66
homology, 73 homotopy, 70
presentation, 67 symmetry, 44
handle, 63 handle slide, 165
Hauptvermutung, 76 Hausdorff, 20
Hessian, 86 homeomorphism, 19
homogeneous, 58 homologous, 74
homology
2-manifolds, 74 coefficients, 79
group, see group, homology persistent, 97
Z
2
, 82 homomorphism, 47
homotopy, 36 equivalence, 36
group, see group, homotopy image, 15
immersion, 22 improper subset, 15
independent, 23, 107 index, 87
induced operation, 46 induced topology, 18
injective, 15 integral domain, 56
integral line, 88 interior, 16
intersection, 15 invariant, 61
irreducible, 56 isomorphism
groups, 48 simplicial complexes, 26
junction, 162 k
-triangle theorem, 104, 130 kernel ker, 48
Klein bottle, 61 knot, 117
linear combination, 23 link, 29, 117
link diagram, 117 linking number
graph, 120 link, 118
loop, 66 manifold, 19–23
connected sum, 62 product, 68
smooth, 21 topological, 20
maximum, 87 merging, 109
metric, 17 metric space, see space, metric