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A Generative Theory of Shape - ( Lecture Notes in Computer Science #2145 )    :  Questioning Klein's Erlanger Program - Michael Leyton

A Generative Theory of Shape - ( Lecture Notes in Computer Science #2145 )

Questioning Klein's Erlanger Program

Paperback Published: October 2006
ISBN: 9783540427179
Number Of Pages: 549

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The purpose of the book is to develop a generative theory of shape that has two properties regarded as fundamental to intelligence - maximizing transfer of structure and maximizing recoverability of the generative operations. These two properties are particularly important in the representation of complex shape - which is the main concern of the book. The primary goal of the theory is the conversion of complexity into understandability. For this purpose, a mathematical theory is presented of how understandability is created in a structure. This is achieved by developing a group-theoretic approach to formalizing transfer and recoverability. To handle complex shape, a new class of groups is developed, called unfolding groups. These unfold structure from a maximally collapsed version of itself. A principal aspect of the theory is that it develops a group-theoretic formalization of major object-oriented concepts such as inheritance. The result is a mathematical language that brings interoperability into the very foundations of geometry. The book gives extensive applications of the theory to CAD/CAM, human and machine vision, robotics, software engineering, and physics.In CAD, lengthy chapters are presented on mechanical and architectural design. For example, using the theory of unfolding groups, the book works in detail through the main stages of mechanical CAD/CAM: part-design, assembly and machining. And within part-design, an extensive algebraic analysis is given of sketching, alignment, dimensioning, resolution, editing, sweeping, feature-addition, and intent-management. In robotics, several levels of analysis are developed for manipulator structure and kinematics. In software, a new theory is given of the principal factors such as text and class structure, object creation and modification, as well as inheritance and hierarchy prediction. In physics, a new theory is given of the conservation laws, and motion decomposition theorems in classical and quantum mechanics.

From the reviews:

"This book is intended for a general scientifically interested audience a ] . The author develops a generative theory of shape along two principles fundamental to intelligence a" maximization of transfer and maximization of recoverability. He proceeds by using an algebraically flavoured approach characterizing features as symmetry groups while the addition of features corresponds to group extension. a ] The generative theory is used in several application areas like visual perception, robotics and computer-aided geometric design." (GA1/4nter Landsman, Zentralblatt MATH, Vol. 1012, 2003)

Transferp. 1
Introductionp. 1
Complex Shape Generationp. 2
Object-Oriented Theory of Geometryp. 3
Transferp. 4
Human Perceptionp. 6
Serial-Link Manipulatorsp. 18
Object-Oriented Inheritancep. 20
Complex Shape Generationp. 21
Designp. 21
Cognition and Transferp. 22
Transfer in Differential Equationsp. 23
Scientific Structurep. 24
Maximization of Transferp. 28
Primitive-Breakingp. 28
The Algebraic Description of Machinesp. 30
Agent Self-Substitutionp. 30
RigorousDefinition of Shapep. 32
RigorousDefinition of Aestheticsp. 32
Shape Generation by Group Extensionsp. 34
Recoverabilityp. 35
Geometry and Memoryp. 35
Practical Need for Recoverabilityp. 35
Theoretical Need for Recoverabilityp. 37
Data Setsp. 37
The Fundamental Recovery Rulesp. 39
Design as Symmetry-Breakingp. 44
Computational Vision and Symmetry-Breakingp. 45
Occupancyp. 47
External vs. Internal Inferencep. 48
Exhaustiveness and Internal Inferencep. 49
Externalization Principlep. 52
Choice of Metric in the Externalization Principlep. 58
Externalization Principle and Environmental Dimensionalityp. 59
History Symmetrization Principlep. 61
Symmetry-to-Trace Conversionp. 63
Rootsp. 66
Inferred Order of the Generative Operationsp. 67
Symmetry-Breaking vs. Asymmetry-Buildingp. 69
Against the Erlanger Programp. 71
Memoryp. 72
Regularityp. 73
Aestheticsp. 74
The Definition of Shapep. 75
Mathematical Theory of Transfer, Ip. 77
Shape Generation by Group Extensionsp. 77
The Importance of n-Cubes in Computational Vision and CADp. 78
Stage 1: Defining Fibers and Controlp. 80
Stage 2: Defining the Fiber-Group Productp. 82
The Fiber-Group Product as a Symmetry Groupp. 84
Defining the Action of the Fiber-Group Product on the Data Set 84
Stage 3: Action of G(C) on the Fiber-Group Productp. 88
Transfer as an Automorphism Groupp. 89
Stage 4: Splitting Extension of the Fiber-Group Product by the Control Groupp. 90
Wreath Productsp. 91
The Universal Embdedding Theoremsp. 93
Nesting vs. Control-Nestingp. 93
Stage 5: Defining the Action of G(F)$$G(C)on F x Cp. 95
Control-GroupIndexesp. 97
Up-Keeping Effect of the Transfer Automorphismsp. 98
The Direct vs. the Indirect Representationp. 103
Transfer asConjugationp. 104
Conjugation and Recoverabilityp. 108
Infinite Control Setsp. 109
The Full Structurep. 110
The FiveGroupActionsp. 112
Mathematical Theory of Transfer, IIp. 115
Introductionp. 115
The Iterated Wreath Productp. 115
Opening Upp. 116
The Group TheoryofHierarchicalDetectionp. 118
Control-Nested t-Automorphismsp. 123
The Wreath Modifierp. 128
Iso-Regular Groupsp. 129
Canonical Plansp. 131
Wreath Poly A Groupsp. 132
WreathCoveringp. 133
Theory of Groupingp. 135
Introductionp. 135
Grouping from Wreath Productsp. 135
Grouping as Algebraic Actionp. 139
Generative Crystallographyp. 140
Using the Law of Groupingp. 141
Hierarchical Detection in Groupingp. 145
Perceptual Relationship between Similar Groupingsp. 146
Product Orderingp. 149
Local-to-Global in a Wreath Productp. 152
Perceptual Effect of Inclusion and Omission of Levelsp. 153
Non-iso-regular Groupsp. 156
Robot Manipulatorsp. 161
Three Algebraic Conditionsp. 161
Object-Centered FramesasTransferp. 162
The Serial-Link Manipulatorp. 165
The Full Group of a Serial-Link Manipulatorp. 167
Transfer in the Serial-Link Manipulatorp. 168
The Full Group of a General-Linked Manipulatorp. 170
Semi-Rigid Groupsp. 172
Including Manipulator Shapep. 173
Algebraic Theory of Inheritancep. 175
Inheritancep. 175
Geometric Inheritancep. 176
Theory of Inheritancep. 177
Relating Inheritance Diagrams to Algebrap. 178
Class Inheritancep. 179
Reference Framesp. 185
Reference Objectsp. 185
Non-coordinate-free Geometryp. 186
Processes andPhasesp. 187
TheoryofReference Objectsp. 188
The Necessity of Reference Framesp. 189
Structure of the 2D Reference Framep. 189
Canonical Plan from the 2D Reference Framep. 193
Organizing Role of the Cartesian Reference Framep. 194
Orientation-and-Formp. 195
Cartesian Frame Bundlep. 200
External Actions on Frames: Decompositionp. 203
The3DReferenceFramep. 205
Assigning Triple-Reflection Structures to Surfacesp. 207
ConstructionPlanep. 210
Relative Motionp. 213
Introductionp. 213
Theory of Relative Motionp. 214
Induced Motionp. 218
Inheritance via Extra Framesp. 221
Physicsp. 225
Surface Primitivesp. 229
Defining and Classifying Primitivesp. 229
Level-ContinuousPrimitivesp. 231
Sphere and Torusp. 232
Cylinder and Conep. 233
Level-Discrete Primitivesp. 234
Formulation of Primitives to Maximize Transfer and Recoverabilityp. 237
Externalizationp. 238
Unfolding Groups, Ip. 239
Symmetry Group of a Complex Environmentp. 239
Concatenation asSymmetry-Breakingp. 240
Concatenation asAsymmetry-Buildingp. 242
Serial-Link Manipulators as Telescope Groupsp. 248
Constructive Solid Geometry(CSG)p. 251
Boolean Operations as Symmetry-Breakingp. 252
Boolean Operations as Telescope Groupsp. 253
Spatial Group Equivalence of Boolean Operationsp. 254
Unfolding Groups, IIp. 257
Importance of Selection in Generativityp. 257
Super-Local Unfoldingp. 258
Establishing a Target for Super-Local Unfoldingp. 259
Super-Local Unfolding and Wreath Coveringsp. 263
The Symmetry Group of a Complex Objectp. 268
Exploitation of Existing Structurep. 268
Cross-Hierarchy in Super-Local Unfoldingp. 268
Unfolding Groups, IIIp. 271
Introductionp. 271
Symmetry Group of an Apartmentp. 272
Wreath-Direct Groupsp. 274
Canonical Unfoldingsp. 278
Incorporating the Symmetry of Referentsp. 282
Why Internal Symmetry Groupsp. 284
Base and Subsidiary Alignment Kernelsp. 284
Cloningp. 285
The Inference Structurep. 286
Group Elementsp. 286
Adding the Anomalyp. 288
Adding more Primitivesp. 293
Multi-index Notationp. 295
Symmetry Streamingp. 296
Complex Shape Generationp. 298
Mechanical Design and Manufacturingp. 299
Introductionp. 299
Parametric, Feature-Based, Solid Modelingp. 299
A Generative Theory of Physical Featuresp. 300
Datum Featuresp. 303
Parent-Child Structures as Wreath Productsp. 305
Complex Shape Generationp. 305
Review of Part Designp. 305
Complex Partsp. 311
A Theory of Resolutionp. 311
A Theory of Sketchingp. 318
A Mathematical Theory of the Designer's Mental Analysisp. 318
Constraints and Unfoldingp. 325
Theory of the Sketch Planep. 326
Solidityp. 329
A Comment on Resolutionp. 329
Adding Featuresp. 330
Model Structurep. 334
Intent Managerp. 335
Intent Managers: Gestalt Principlesp. 336
Slicing asUnfoldingp. 336
Assembly: Symmetry-Breaking Theoryp. 338
Unfolding Groups, Boolean Operations, and Assemblyp. 339
The Designer's Conceptual Planningp. 343
Holesthrough Several Layersp. 345
Analogy with Quantum Mechanicsp. 346
Fiber-Relative Actionsp. 346
The Full Group of the Robot Serial-Link Manipulatorp. 353
Machiningp. 357
A Mathematical Theory of Architecturep. 365
Introductionp. 365
The Design Processp. 366
Massing Studiesp. 366
Mass Elementsp. 367
The Hierarchy of Mass Groupsp. 367
Symmetry Group of a Massing Structurep. 368
Massing Structure and Generativityp. 374
Slicing the Massing Study to Create Floorplatesp. 375
Space Planning: Unfolding of Space Volumesp. 378
Space Planning: Unfolding the Boundary and Void Spacesp. 379
Unfolding the Room Volumesp. 380
Unfolding the Wall Structurep. 380
Complex Slicingp. 381
Design Development Phasep. 383
Choice of Materialsp. 383
Doorsand Windowsp. 384
Structural Column Gridp. 386
Ceiling Gridp. 388
Stairsp. 388
Shaftsp. 389
Roofp. 390
Development of Accuracyp. 390
Construction Documentsp. 391
Sectionsand Elevationsp. 391
Conclusionp. 393
Summary of a Mathematical Theory of Architecturep. 393
Solid Structurep. 397
Introductionp. 397
The Solid Primitivesp. 397
The Solid n-Cubep. 398
The Hyperoctahedral Wreath Hyperplane Groupp. 398
CubesasCartesian Framesp. 400
The Symmetry Group of the Solid n-Cubep. 402
Solid Interval and Solid Squarep. 409
The Other Solid Primitivesp. 411
The Solid Spherep. 413
The Solid Cross-Section Cylinderp. 413
The Solid Ruled Cylinderp. 415
TheSolidCross-SectionBlockp. 416
TheSolidRuledorPlanarBlockp. 417
The Full Set of Solid Primitivesp. 417
Externalization in the Solid Primitivesp. 418
The Unfolding Group of a Solidp. 420
Wreath Formulation of Splinesp. 423
TheGoalofThisChapterp. 423
CurvesasMachinesp. 428
Cubic Hermite Curvesp. 429
Parametrized SurfacesasMachinesp. 432
Bicubic Hermite Surfacesp. 433
Parametrized 3-SolidsasMachinesp. 436
Tricubic Hermite Solidp. 438
Final Commentp. 440
Wreath Formulation of Sweep Representationsp. 443
Sweep Representationsp. 443
Aesthetics and Sweep Representationsp. 449
Ray Representationsp. 450
Multiple Sweepingp. 453
Process Grammarp. 455
Introductionp. 455
Inference from a Single Shapep. 455
Intervening Historyp. 461
Process Grammarp. 463
Other Literaturep. 466
Conservation Laws of Physicsp. 467
Wreath Products and Commutatorsp. 467
Transfer in Quantum Mechanicsp. 469
Symmetriesof the Schrodinger Equationp. 469
Space-Time Transfer in Quantum Mechanicsp. 470
Non-solvability of the Galilean Lie Algebrap. 473
Semisimple Lie Algebras in Quantum Mechanicsp. 474
Musicp. 477
Introductionp. 477
Motival Materialp. 477
Modulation as a Wreath Productp. 478
Psychological Studies of Sequential Structurep. 479
Transfer in Musical Sequence Structurep. 480
Meterp. 481
AlgebraicTheoryofMeterp. 485
TheoryofMetricalMovement(Pulse)p. 489
Algebraic Structure of Groupingp. 491
Against the Erlanger Programp. 495
Introductionp. 495
Orientation-and-Formp. 497
The Generative Structure of Quadrilateralsp. 497
Non-coordinate-freedomp. 503
Theorem-Proving in Geometryp. 506
The Geometry Hierarchyp. 507
Projective Asymmetrization: Extrinsic Viewp. 509
Deriving Projective CoordinateSystemsp. 513
Non-transitivity of the Geometry Groupp. 517
Regular Translation Structurep. 518
3D ProjectiveAsymmetrizationp. 519
Against the Erlanger Program: Summaryp. 526
Semi-direct Productsp. 531
Normal Subgroupsp. 531
Semi-direct Productsp. 533
The Extending Group H as an Automorphism Groupp. 535
Multiplication in a Semi-direct Productp. 536
Direct Productsp. 537
Symbolsp. 539
Referencesp. 541
Indexp. 549
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9783540427179
ISBN-10: 3540427171
Series: Lecture Notes in Computer Science
Audience: General
Format: Paperback
Language: English
Number Of Pages: 549
Published: October 2006
Publisher: Springer-Verlag Berlin and Heidelberg Gmbh & Co. Kg
Country of Publication: DE
Dimensions (cm): 23.39 x 15.6  x 2.97
Weight (kg): 0.79