A Generative Theory of Shape
Michael Leyton
Summary:
This talk
gives an introduction to my book, A Generative Theory of Shape
(Springer-Verlag, 550pages). 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 that structure. 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 an object-oriented theory of geometry.
The
algebraic theory is applied in detail to CAD, perception, and robotics. 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. The equivalent analysis is also done for architectural
design. In perception, extensive theories are given for grouping and the main
Gestalt motion phenomena (induced motion, separation of systems, the Johannson
relative/absolute motion effects); as well as orientation and form. In
robotics, several levels of analysis are developed for manipulator structure,
using the author’s algebraic theory of object-oriented structure.