Overview

Completely updated to include the most recent developments in the field, the third edition like the two previous editions, emphasizes clarity and thoroughness in the mathematical development of its subjects. It is written in a style that is free of jargon of special applications, while integrating the three important functions of geometric modeling: to represent elementary forms (curves, surfaces, and solids), to shape and assemble these into complex forms, and to determine geometric properties and relationships. With hundreds of illustrations, this unique book appeals to the readers visual and intuitive skills in a way that makes it easier to understand its more abstract concepts. Upper-division and graduate students, teachers, and professionals studying, teaching or practicing geometric modeling, 3D modeling, computational geometry, computer graphics applications, animation, CAD/CAM, and related subjects will find this to be a very valuable reference.

 

Table of  Contents

Introduction

   What is Geometric Modeling?

   History

   The Mathematics

   Vectors

   Matrices

   Conventions and Notation

   

Curves

   Intrinsic Equations of Curves

   Explicit and Implicit Equations of Curves

   Parametric Equations of Curves

   Conic Curves

   Points on a Curve

   Hermite, Bezier, and B-Spline Curves: An Overview

 

Hermite Curves

   Algebraic and Geomtric Forms

   Hermite Basis Functions

   Matrix Form

   Tangent Vectors

   Truncating and Subdividing

   Three-Point Interpolation

   Four-Point Interpolation

   Conic Hermite Curves

   Composite Hermite Curve

 

Bezier Curves

   Bezier Basis Functions

   Control Points

   Truncating and Subdividing

   Composite Bezier Curves

   Rational Bezier Curves

   

B-Spline Curves

   Nonuniform B-Spline Basis Functions

   Uniform B-Spline Basis Functions

   Quadratic and Cubic B-Spline Basis Functions

   Closed B-Spline Curves

   Continuity

   Conversion between Basis Functions

   Nonuniform Rational B-Spline Curves

   Representing Conics with NURBS Curves

   Cubic Beta Splines

 

Surfaces  

   Explicit and Implicit Equations of Surfaces

   Quadric Surfaces

   Parametric Equation of Surfaces

   Points on a Surface

   Curve Nets

   Embedded Curves

   Hermite, Bezier, and B-Spline Surfaces: An Overview

 

 Bicubic Hermite Surfaces

   Algebraic and Geometric Forms

   Hermite Patch Basis Functions

   Tangent and Twist Vectors

   Normals

   16-Point Form

   Reparameterization of a Patch

   Truncating and Subdividing a Patch

   Composite Hermite Surfaces

   Special Hermite Patches

   Bend Surfaces

 

Bezier Surfaces

   The Tensor Product Bezier Patch

   The Bicubic Bezier Patch

   A 3 x 5 Rectangular Array of Control Points

   Converting between Bicubic Bezier and Hermite Forms

   Degree Elevation in a Bezier Surface

   Composite Bezier Surfaces

   Rational Bezier Patch

 

B-Spline Surfaces

   The Tensor Product B-Spline Surface

   Matrix Form

   Open and Closed B-Spline Surfaces

   Nonuniform Rational B-Spline Surfaces

 

Solids

   Parametric Solids

   The Tricubic Solid

   Curves and Surfaces Embedded in a Solid

   Generalized Notation Scheme and Higher-Dimension Elements

   Instances and Parameterized Shapes

   Sweep Solids

   Controlled Deformation Solids

 

Complex Model Construction

   Topology of Models

   Graph-Based Models

   Boolean Models

   Boolean Model Construction

   Constructive Solid Geometry

   Boundary Models

 

Geometric Properties

   Local Properties of a Curve

   Global Properties of a Curve

   Global Properties of a Surface

   Global Properties of Complex Solids

   Relational Properties

   Intersections

 

Answers to Selected Exercises

 

Index