CAD: Computational Concepts and Methods, Softcover reprint of the original 1st ed. 1986 computational concepts and methods

Intersection of straight line segments 103 Non-linearity and the intersection of curves 109 Subdivision and box-testing techniques 114 Closed curves 120 A data structure for closed curves 123 The merging of closed curves 125 Chapter 5: Representation of surfaces 133 Introduction 133 Ruled surfaces 134 Coons' patches 136 Surfaces through given points 141 Bezier surface patches 143 B-spline surfaces 148 The DUCT system 149 Problems involved in putting patches together 153 Chapter 6: MODCON: an example system 159 Background to the system 159 The use of primitive shapes 160 Putting primitives together 165 A simple example 170 Operation of the system 171 Limitations of the system 174 Conclusions 176 Chapter 7: Introduction to surface and solid modelling 177 Introduction 177 Types of geometric modeller 177 Solid modelling 181 Obtaining volumetric properties 186 Defining primitive volumetric shapes 188 Hidden line removal and surface shading 190 References 197 Index 199 CHAPTER 1 Review of some basic ideas Introduction The purpose of this chapter is to review some of the standard notation and concepts that underlie the material to be presented later. These are the ideas of mathematics and of numerical mathematics. They are important from a computer­ aided design (CAD) point of view because the only way in which we can persuade a computer system to deal with geometry for us is by reducing it to a set of numbers which can then be stored and manipulated.

1: Review of some basic ideas.- Cartesian coordinates.- Use of vectors.- Some simple geometric shapes.- Use of matrix notation.- The standard linear equation.- Non-linear equations.- 2: Simple datastructures and transformations.- Use of nodes.- Node and edge lists.- Transformations of the nodes.- Three-dimensional coordinates.- Transformations in three dimensions.- Homogeneous coordinates.- Homogeneous transformation matrices.- View transformations.- Application to assemblies of shapes.- Application to robotics.- 3: Representation of curves.- Implicit equations.- Use of parameters.- The Bézier formulation.- More general Bézier forms.- Manipulating Bézier segments.- Bézier segments with homogeneous coordinates.- The de Casteljau algorithm.- to B-spline curves.- B-spline basis functions.- B-splines and some of their properties.- Extensions of B-spline ideas.- Joining curve segments together.- Curves through given points.- 4: Intersection of curves.- Intersection of straight line segments.- Non-linearity and the intersection of curves.- Subdivision and box-testing techniques.- Closed curves.- A data structure for closed curves.- The merging of closed curves.- 5: Representation of surfaces.- Ruled surfaces.- Coons’ patches.- Surfaces through given points.- Bézier surface patches.- B-spline surfaces.- The DUCT system.- Problems involved in putting patches together.- 6: MODCON: an example system.- Background to the system.- The use of primitive shapes.- Putting primitives together.- A simple example.- Operation of the system.- Limitations of the system.- Conclusions.- 7: Introduction to surface and solid modelling.- Types of geometric modeller.- Solid modelling.- Obtaining volumetric properties.- Defining primitive volumetric shapes.- Hidden line removal and surface shading.- References.