1st Edition

# Theory of Dimensioning An Introduction to Parameterizing Geometric Models

350 Pages
by CRC Press

Encompassing a wide range of mathematical concepts, this text/reference presents a comprehensive theory of dimensioning and parameterizing of geometric models. This volume develops a unified and systematic theory of intrinsic and relational dimensioning using the powerful notion of congruence.

Packed with illustrative examples and exercises, it explains how basic geometric knowledge can be used to understand and approach various dimensioning challenges and provides valuable methods for parameterizing geometric models. This valuable reference discusses how dimensional constraints are resolved and managed and offers effective techniques to dimension and parameterize solids.

Introduction
An Earlier Theory of Dimensioning
Outline of a Modern Theory of Dimensioning
Standardized Indication of Dimensions
Exercises
Notes and References
Congruence
Point-Sets and Tuples
Rigid Motion
Reflexive Symmetry and Chirality
Vectors and Oriented Point-Sets
Congruence, Dimensions, and Parameters
Exercises
Notes and References
Dimensioning Elementary Curves
Conics
Free-Form Curves
Space Curves
Exercises
Notes and References
Dimensioning Elementary Surfaces
Free-Form Surfaces
Swept Surfaces
Exercises
Notes and References
Dimensioning Relative Positions of Elementary Objects
Distances and Angles
Some Cases Involving Points
Relative Positioning Two Lines
Relative Positioning a Line and a Plane
Relative Positioning Two Planes
Cases Involving Oriented Lines and Oriented Planes
Cases Involving Helices
Summary
Exercises
Notes and References
Symmetry
Groups
Symmetry Groups
Connected Lie Subgroups of the Rigid Motion Group
Classification of Continuous Symmetry Groups
Classification of Tuples of Sets
Classification of Lower-Order Kinematic Pairs
Summary
Exercises
Notes and References
General Theory of Dimensioning Relative Positions
Tuple Congruence
Number of Dimensions for Relative Positions
More on Relative Positioning Spherical, Cylindrical, Planar, and Helical Classes
Summary
Exercises
Notes and References
Dimensional Constraints
Basic Geometric Constraints
Rigidity Theory
Inducing Hierarchy in Simultaneity
Summary
Exercises
Notes and References
Dimensioning Solids
Dimensioning a Solid Polyhedron
Dimensioning Procedurally Defined Solids
Dimensioning Features
Summary
Exercises
Notes and References
Appendix 1: Matrices
Appendix 2: Groups
Appendix 3: Graphs
Appendix 4: Solids
Bibliography
Index

### Biography

Vijay Srinivasan

“This textbook provides a rigorous mathematical foundation for the common engineering activity of dimensioning and tolerancing in the context of computer-aided design CAD. The mathematical prerequisites are basic geometry and linear algebra. … appendices meant to help the engineering reader to which it is addressed, provide information on the mathematics involved.”
— Victor V. Pambuccian (Phoenix), in Zentralblatt MATH, Vol. 1102, 2007/04
“…a ‘status report’ from the research frontiers….clearly and gracefully written, with many examples and exercises, it will almost certainly be used as a university text, and as a standard source for researchers.”
Herbert B. Voelcker, Professor, Cornell University, Ithaca, New York, USA
“…fills a major void in the engineering literature.”
Michael J. Pratt, LMR Systems, England
“…an extremely impressive unification of the conceptual and mathematical foundations of dimensioning….I strongly recommend this book to both students and advanced experts.  It is a true milestone.”
Michael Leyton, Professor, Rutgers University, New Brunswick, New Jersey, USA