Geometric folding algorithms: linkages, origami, polyhedra
Folding and unfolding problems have been implicit since Albrecht Dürer in the early 1500s, but have only recently been studied in the mathematical literature. Emphasising algorithmic or computational aspects, this treatment of the geometry of folding and unfolding presents over 60 unsolved 'ope...
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge [u.a.]
Cambridge Univ. Press
2007
|
| Edition: | 1. publ. |
| Subjects: | |
| Online Access: | Publisher description Table of contents only Inhaltsverzeichnis |
| Summary: | Folding and unfolding problems have been implicit since Albrecht Dürer in the early 1500s, but have only recently been studied in the mathematical literature. Emphasising algorithmic or computational aspects, this treatment of the geometry of folding and unfolding presents over 60 unsolved 'open problems' to spur further research. |
| Item Description: | Includes index. |
| Physical Description: | XIII, 472 S. zahlr. graph. Darst. |
| ISBN: | 9780521857574 0521857570 |
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| 520 | 3 | |a Folding and unfolding problems have been implicit since Albrecht Dürer in the early 1500s, but have only recently been studied in the mathematical literature. Emphasising algorithmic or computational aspects, this treatment of the geometry of folding and unfolding presents over 60 unsolved 'open problems' to spur further research. | |
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Record in the Search Index
| _version_ | 1804136779855429632 |
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| adam_text | Titel: Geometric folding algorithms
Autor: Demaine, Erik D.
Jahr: 2007
Contents
Preface page xi
0 Introduction 1
0.1 Design Problems 1
0.2 Foldability Questions 3
Part I. Linkages
1 Problem Classification and Examples 9
1.1 Classification 10
1.2 Applications 11
2 Upper and Lower Bounds 17
2.1 General Algorithms and Upper Bounds 17
2.2 Lower Bounds 22
3 Planar Linkage Mechanisms 29
3.1 Straight-Line Linkages 29
3.2 Kempe s Universality Theorem 31
3.3 Hart s Inversor 40
4 Rigid Frameworks 43
4.1 Brief History 43
4.2 Rigidity 43
4.3 Generic Rigidity 44
4.4 Infinitesimal Rigidity 49
4.5 Tensegrities 53
4.6 Polyhedral Liftings 57
5 Reconfiguration of Chains 59
5.1 Reconfiguration Permitting Intersection 59
5.2 Reconfiguration in Confined Regions 67
5.3 Reconfiguration Without Self-Crossing 70
6 Locked Chains 86
6.1 Introduction 86
6.2 History 87
vii
^VHraH I Contents !
6.3 Locked Chains in 3D 88
6.4 No Locked Chains in 4D 92
6.5 Locked Trees in 2D 94
6.6 No Locked Chains in 2D 96
6.7 Algorithms for Unlocking 2D Chains 105
6.8 Infinitesimally Locked Linkages in 2D 113
6.9 3D Polygons with a Simple Projection 119
7 Interlocked Chains 123
7.1 2-Chains 125
7.2 3-Chains 126
7.3 4-Chains 127
8 Joint-Constrained Motion 131
8.1 Fixed-Angle Linkages 131
8.2 Convex Chains 143
9 Protein Folding 148
9.1 Producible Polygonal Protein Chains 148
9.2 Probabilistic Roadmaps 154
9.3 HP Model 158
Part II. Paper
10 Introduction 167
10.1 History of Origami 167
10.2 History of Origami Mathematics 168
10.3 Terminology 169
10.4 Overview 170
11 Foundations 172
11.1 Definitions: Getting Started 172
11.2 Definitions: Folded States of ID Paper 175
11.3 Definitions: Folding Motions of ID Paper 182
11.4 Definitions: Folded States of 2D Paper 183
11.5 Definitions: Folding Motions of 2D Paper 187
11.6 Folding Motions Exist 189
12 Simple Crease Patterns 193
12.1 One-Dimensional Flat Foldings 193
12.2 Single-Vertex Crease Patterns 198
12.3 Continuous Single-Vertex Foldability 212
13 General Crease Patterns 214
13.1 Local Flat Foldability is Easy 214
13.2 Global Flat Foldability is Hard 217
14 MapFoldlng 224
14.1 Simple Folds 225
14.2 Rectangular Maps: Reduction to ID 227
14.3 Hardness of Folding Orthogonal Polygons 228
14.4 Open Problems 230
I
Contents I HVEHJ
15 Silhouettes and Gift Wrapping 232
15.1 Strip Folding 233
15.2 Hamiltonian Triangulation 233
15.3 Seam Placement 236
15.4 Efficient Foldings 237
16 The Tree Method 240
16.1 Origami Bases 240
16.2 Uniaxial Bases 242
16.3 Everything is Possible 243
16.4 Active Paths 244
16.5 Scale Optimization 246
16.6 Convex Decomposition 247
16.7 Overview of Folding 249
16.8 Universal Molecule 250
17 One Complete Straight Cut 254
17.1 Straight-Skeleton Method 256
17.2 Disk-Packing Method 263
18 Flattening Polyhedra 279
18.1 Connection to Part III: Models of Folding 279
18.2 Connection to Fold-and-Cut Problem 280
18.3 Solution via Disk Packing 281
18.4 Partial Solution via Straight Skeleton 281
19 Geometric Constructibility 285
19.1 Trisection 285
19.2 Huzita s Axioms and Hatori s Addition 285
19.3 Constructible Numbers 288
19.4 Folding Regular Polygons 289
19.5 Generalizing the Axioms to Solve All Polynomials? 290
20 Rigid Origami and Curved Creases 292
20.1 Folding Paper Bags 292
20.2 Curved Surface Approximation 293
20.3 David Huffman s Curved-Folds Origami 296
Part III. Polyhedra
21 Introduction and Overview . , 299
21.1 Overview 299
21.2 Curvature 301
21.3 Gauss-Bonnet Theorem 304
22 Edge Unfolding of Polyhedra 306
22.1 Introduction 306
; 22.2 Evidence for Edge Unfoldings 312
22.3 Evidence against Edge Unfoldings 313
22.4 Unfoldable Polyhedra 318
22.5 Special Classes of Edge-Unfoldable Polyhedra 321
22.6 Vertex Unfoldings 333
^IQJH Contents
23 Reconstruction of Polyhedra 339
23.1 Cauchy s Rigidity Theorem 341
23.2 Flexible Polyhedra 345
23.3 Alexandrov s Theorem 348
23.4 Sabitov s Algorithm 354
24 Shortest Paths and Geodesies 358
24.1 Introduction 358
24.2 Shortest Paths Algorithms 362
24.3 Star Unfolding 366
24.4 Geodesies: Lyustermk-Schnirelmann 372
24.5 Curve Development 375
25 Folding Polygons to Polyhedra 381
25.1 Folding Polygons: Preliminaries 381
25.2 Edge-to-Edge Gluings 386
25.3 Gluing Trees 392
- 25.4 Exponential Number of Gluing Trees 396
25.5 General Gluing Algorithm 399
25.6 The Foldings of the Latin Cross 402
25.7 The Foldings of a Square to Convex Polyhedra 411
25.8 Consequences and Conjectures 418
25.9 Enumerations of Foldings 426
25.10 Enumerations of Cuttings 429
25.11 Orthogonal Polyhedra 431
26 Higher Dimensions 437
26.1 Parti 437
26.2 Part II 437
26.3 Part III 438
Bibliography 443
Index 461
|
| adam_txt |
Titel: Geometric folding algorithms
Autor: Demaine, Erik D.
Jahr: 2007
Contents
Preface page xi
0 Introduction 1
0.1 Design Problems 1
0.2 Foldability Questions 3
Part I. Linkages
1 Problem Classification and Examples 9
1.1 Classification 10
1.2 Applications 11
2 Upper and Lower Bounds 17
2.1 General Algorithms and Upper Bounds 17
2.2 Lower Bounds 22
3 Planar Linkage Mechanisms 29
3.1 Straight-Line Linkages 29
3.2 Kempe's Universality Theorem 31
3.3 Hart's Inversor 40
4 Rigid Frameworks 43
4.1 Brief History 43
4.2 Rigidity 43
4.3 Generic Rigidity 44
4.4 Infinitesimal Rigidity 49
4.5 Tensegrities 53
4.6 Polyhedral Liftings 57
5 Reconfiguration of Chains 59
5.1 Reconfiguration Permitting Intersection 59
5.2 Reconfiguration in Confined Regions 67
5.3 Reconfiguration Without Self-Crossing 70
6 Locked Chains 86
6.1 Introduction 86
6.2 History 87
vii
^VHraH I Contents !
6.3 Locked Chains in 3D 88
6.4 No Locked Chains in 4D 92
6.5 Locked Trees in 2D 94
6.6 No Locked Chains in 2D 96
6.7 Algorithms for Unlocking 2D Chains 105
6.8 Infinitesimally Locked Linkages in 2D 113
6.9 3D Polygons with a Simple Projection 119
7 Interlocked Chains 123
7.1 2-Chains 125
7.2 3-Chains 126
7.3 4-Chains 127
8 Joint-Constrained Motion 131
8.1 Fixed-Angle Linkages 131
8.2 Convex Chains 143
9 Protein Folding 148
9.1 Producible Polygonal Protein Chains 148
9.2 Probabilistic Roadmaps 154
9.3 HP Model 158
Part II. Paper
10 Introduction 167
10.1 History of Origami 167
10.2 History of Origami Mathematics 168
10.3 Terminology 169
10.4 Overview 170
11 Foundations 172
11.1 Definitions: Getting Started 172
11.2 Definitions: Folded States of ID Paper 175
11.3 Definitions: Folding Motions of ID Paper 182
11.4 Definitions: Folded States of 2D Paper 183
11.5 Definitions: Folding Motions of 2D Paper 187
11.6 Folding Motions Exist 189
12 Simple Crease Patterns 193
12.1 One-Dimensional Flat Foldings 193
12.2 Single-Vertex Crease Patterns 198
12.3 Continuous Single-Vertex Foldability 212
13 General Crease Patterns 214
13.1 Local Flat Foldability is Easy 214
13.2 Global Flat Foldability is Hard 217
14 MapFoldlng 224
14.1 Simple Folds 225
14.2 Rectangular Maps: Reduction to ID 227
14.3 Hardness of Folding Orthogonal Polygons 228
14.4 Open Problems 230
I
Contents I HVEHJ
15 Silhouettes and Gift Wrapping 232
15.1 Strip Folding 233
15.2 Hamiltonian Triangulation 233
15.3 Seam Placement 236
15.4 Efficient Foldings 237
16 The Tree Method 240
16.1 Origami Bases 240
16.2 Uniaxial Bases 242
16.3 Everything is Possible 243
16.4 Active Paths 244
16.5 Scale Optimization 246
16.6 Convex Decomposition 247
16.7 Overview of Folding 249
16.8 Universal Molecule 250
17 One Complete Straight Cut 254
17.1 Straight-Skeleton Method 256
17.2 Disk-Packing Method 263
18 Flattening Polyhedra 279
18.1 Connection to Part III: Models of Folding 279
18.2 Connection to Fold-and-Cut Problem 280
18.3 Solution via Disk Packing 281
18.4 Partial Solution via Straight Skeleton 281
19 Geometric Constructibility 285
19.1 Trisection 285
19.2 Huzita's Axioms and Hatori's Addition 285
19.3 Constructible Numbers 288
19.4 Folding Regular Polygons 289
19.5 Generalizing the Axioms to Solve All Polynomials? 290
20 Rigid Origami and Curved Creases 292
20.1 Folding Paper Bags 292
20.2 Curved Surface Approximation 293
20.3 David Huffman's Curved-Folds Origami 296
Part III. Polyhedra
21 Introduction and Overview . , 299
21.1 Overview 299
21.2 Curvature 301
21.3 Gauss-Bonnet Theorem 304
22 Edge Unfolding of Polyhedra 306
22.1 Introduction 306
; 22.2 Evidence for Edge Unfoldings 312
22.3 Evidence against Edge Unfoldings 313
22.4 Unfoldable Polyhedra 318
22.5 Special Classes of Edge-Unfoldable Polyhedra 321
22.6 Vertex Unfoldings 333
^IQJH Contents
23 Reconstruction of Polyhedra 339
23.1 Cauchy's Rigidity Theorem 341
23.2 Flexible Polyhedra 345
23.3 Alexandrov's Theorem 348
23.4 Sabitov's Algorithm 354
24 Shortest Paths and Geodesies 358
24.1 Introduction 358
24.2 Shortest Paths Algorithms 362
24.3 Star Unfolding 366
24.4 Geodesies: Lyustermk-Schnirelmann 372
24.5 Curve Development 375
25 Folding Polygons to Polyhedra 381
25.1 Folding Polygons: Preliminaries 381
25.2 Edge-to-Edge Gluings 386
25.3 Gluing Trees 392
- 25.4 Exponential Number of Gluing Trees 396
25.5 General Gluing Algorithm 399
25.6 The Foldings of the Latin Cross 402
25.7 The Foldings of a Square to Convex Polyhedra 411
25.8 Consequences and Conjectures 418
25.9 Enumerations of Foldings 426
25.10 Enumerations of Cuttings 429
25.11 Orthogonal Polyhedra 431
26 Higher Dimensions 437
26.1 Parti 437
26.2 Part II 437
26.3 Part III 438
Bibliography 443
Index 461 |
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| publisher | Cambridge Univ. Press |
| record_format | marc |
| spelling | Demaine, Erik D. 1981- Verfasser (DE-588)123829704 aut Geometric folding algorithms linkages, origami, polyhedra Erik D. Demaine, Joseph O'Rourke 1. publ. Cambridge [u.a.] Cambridge Univ. Press 2007 XIII, 472 S. zahlr. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Includes index. Folding and unfolding problems have been implicit since Albrecht Dürer in the early 1500s, but have only recently been studied in the mathematical literature. Emphasising algorithmic or computational aspects, this treatment of the geometry of folding and unfolding presents over 60 unsolved 'open problems' to spur further research. Polyèdres - Informatique Polyèdres - Modèles Datenverarbeitung Polyhedra Models Polyhedra Data processing Polyeder (DE-588)4132101-7 gnd rswk-swf Falten (DE-588)4153638-1 gnd rswk-swf Geometrie (DE-588)4020236-7 gnd rswk-swf Polyeder (DE-588)4132101-7 s Falten (DE-588)4153638-1 s Geometrie (DE-588)4020236-7 s DE-604 O'Rourke, Joseph 1951- Verfasser (DE-588)114449708 aut http://www.loc.gov/catdir/enhancements/fy0703/2006038156-d.html Publisher description http://www.loc.gov/catdir/enhancements/fy0703/2006038156-t.html Table of contents only HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015826302&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Demaine, Erik D. 1981- O'Rourke, Joseph 1951- Geometric folding algorithms linkages, origami, polyhedra Polyèdres - Informatique Polyèdres - Modèles Datenverarbeitung Polyhedra Models Polyhedra Data processing Polyeder (DE-588)4132101-7 gnd Falten (DE-588)4153638-1 gnd Geometrie (DE-588)4020236-7 gnd |
| subject_GND | (DE-588)4132101-7 (DE-588)4153638-1 (DE-588)4020236-7 |
| title | Geometric folding algorithms linkages, origami, polyhedra |
| title_auth | Geometric folding algorithms linkages, origami, polyhedra |
| title_exact_search | Geometric folding algorithms linkages, origami, polyhedra |
| title_exact_search_txtP | Geometric folding algorithms linkages, origami, polyhedra |
| title_full | Geometric folding algorithms linkages, origami, polyhedra Erik D. Demaine, Joseph O'Rourke |
| title_fullStr | Geometric folding algorithms linkages, origami, polyhedra Erik D. Demaine, Joseph O'Rourke |
| title_full_unstemmed | Geometric folding algorithms linkages, origami, polyhedra Erik D. Demaine, Joseph O'Rourke |
| title_short | Geometric folding algorithms |
| title_sort | geometric folding algorithms linkages origami polyhedra |
| title_sub | linkages, origami, polyhedra |
| topic | Polyèdres - Informatique Polyèdres - Modèles Datenverarbeitung Polyhedra Models Polyhedra Data processing Polyeder (DE-588)4132101-7 gnd Falten (DE-588)4153638-1 gnd Geometrie (DE-588)4020236-7 gnd |
| topic_facet | Polyèdres - Informatique Polyèdres - Modèles Datenverarbeitung Polyhedra Models Polyhedra Data processing Polyeder Falten Geometrie |
| url | http://www.loc.gov/catdir/enhancements/fy0703/2006038156-d.html http://www.loc.gov/catdir/enhancements/fy0703/2006038156-t.html http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015826302&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT demaineerikd geometricfoldingalgorithmslinkagesorigamipolyhedra AT orourkejoseph geometricfoldingalgorithmslinkagesorigamipolyhedra |