Computer-Aided Geometric Design: A Totally Four-Dimensional Approach

✍ Scribed by Fujio Yamaguchi (auth.)

• Publisher: Springer Tokyo
• Year: 2002
• Tongue: English
• Leaves: 644
• Edition: 1
• Category: Library

✦ Synopsis

Computer graphics, computer-aided design, and computer-aided manufacturing are tools that have become indispensable to a wide array of activities in contemporary society. Euclidean processing provides the basis for these computer-aided design systems although it contains elements that inevitably lead to an inaccurate, non-robust, and complex system. The primary cause of the deficiencies of Euclidean processing is the division operation, which becomes necessary if an n-space problem is to be processed in n-space. The difficulties that accompany the division operation may be avoided if processing is conducted entirely in (n+1)-space. The paradigm attained through the logical extension of this approach, totally four-dimensional processing, is the subject of this book. This book offers a new system of geometric processing techniques that attain accurate, robust, and compact computations, and allow the construction of a systematically structured CAD system.

✦ Table of Contents

Front Matter....Pages ii-xxxix
Front Matter....Pages 1-1
Problems of Euclidean Approach....Pages 3-11
Homogeneous Coordinates....Pages 13-20
Classical Projective Space....Pages 21-31
Two-Sided Space....Pages 33-65
Processing in Projective Spaces....Pages 67-69
Transformations....Pages 71-84
Front Matter....Pages 85-85
A Solution....Pages 87-103
Geometric Definitions in General....Pages 105-121
Geometric Element Definitions....Pages 123-158
Homogeneous Curves....Pages 159-198
Homogeneous Surfaces....Pages 199-206
Basic Intersections....Pages 207-213
4 × 4 Determinant Methods....Pages 215-246
The Homogeneous Determinant Method....Pages 247-297
Geometric Newton-Raphson Method....Pages 299-324
Exact Arithmetic....Pages 325-349
Fundamentals of Topology....Pages 351-380
Data Structures....Pages 381-415
Euclidean and Affine....Pages 417-434
Set Operations....Pages 435-466
Front Matter....Pages 85-85
Polyhedron Subdivisions....Pages 467-487
Reconstruction of 3-D Points....Pages 489-494
Display Process....Pages 495-517
Classical or Two-Sided?....Pages 519-523
Front Matter....Pages 525-525
Various Tests....Pages 527-557
Front Matter....Pages 559-559
Evaluations....Pages 561-569
Role of Division Operations....Pages 571-574
Various Interpretations....Pages 575-577
Back Matter....Pages 579-630

✦ Subjects

Computer-Aided Engineering (CAD, CAE) and Design