A Generative Theory of Shape

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Springer Science & Business Media, Oct 26, 2001 - Computers - 554 pages
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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.
 

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Contents

1 Transfer
1
2 Recoverability
35
3 Mathematical Theory of Transfer I
77
4 Mathematical Theory of Transfer II
115
5 Theory of Grouping
135
6 Robot Manipulators
160
7 Algebraic Theory of Inheritance
175
8 Reference Frames
184
15 A Mathematical Theory of Architecture
365
16 Solid Structure
396
17 Wreath Formulation of Splines
423
18 Wreath Formulation of Sweep Representations
443
19 Process Grammar
455
20 Conservation Laws of Physics
467
21 Music
477
22 Against the Erlanger Program
494

9 Relative Motion
213
10 Surface Primitives
229
11 Unfolding Groups I
239
12 Unfolding Groups II
256
13 Unfolding Groups III
271
14 Mechanical Design and Manufacturing
299
ASemidirect Products
531
BSymbols
539
References
540
Index
549
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About the author (2001)

Michael Leyton is a professor in the Psychology Department at Rutgers University. He is a recipient of the Presidential Young Investigatory Award for outstanding work in cognitive science.