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