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

by Fujio Yamaguchi

Overview

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.

Product Details

ISBN-13:	9784431703402
Publisher:	Springer Japan
Publication date:	11/12/2002
Edition description:	2002
Pages:	630
Product dimensions:	6.10(w) x 9.25(h) x 0.07(d)

Table of Contents

I Problem Definitions.- 1 Problems of Euclidean Approach.- 2 Homogeneous Coordinates.- 3 Classical Projective Space.- 4 Two-Sided Space.- 5 Processing in Projective Spaces.- 6 Transformations.- II Presentation of a Solution-Four-dimensional Processing-Four-dimensional Processing-.- 7 A Solution.- 8 Geometric Definitions in General.- 9 Geometric Element Definitions.- 10 Homogeneous Curves.- 11 Homogeneous Surfaces.- 12 Basic Intersections.- 13 4 × 4 Determinant Methods.- 14 The Homogeneous Determinant Method.- 15 Geometric Newton-Raphson Method.- 16 Exact Arithmetic.- 17 Fundamentals of Topology.- 18 Data Structures.- 19 Euclidean and Affine.- 20 Set Operations.- 21 Polyhedron Subdivisions.- 22 Reconstruction of 3-D Points.- 23 Display Process.- 24 Classical or Two-Sided—.- III Test Results.- 25 Various Tests.- IV Conclusions.- 26 Evaluations.- 27 Role of Division Operations.- 28 Various Interpretations.- Afterword.- A Fundamentals on Euclidean Processing.- A.1.- A.2.- A.3.- A.4.- B Various Geometries.- C Plane Determinant Method.- C.1.- C.2.- C.3.- C.4.- C.5.- C.5.1.- C.5.2.- D Analytical Geometry in Vector Space.- D.1.- D.1.1.- D.1.2.- D.1.3.- D.2.- D.2.1.- E Proofs for Chapter 16.- E.1.- E.2.