An Integrated Introduction to Computer Graphics and Geometric Modeling Book

Two-Dimensional Computer Graphics: From Common Curves to Intricate Fractals

Turtle Graphics

Turtle Graphics

Turtle Commands

Turtle Programs

Fractals from Recursive Turtle Programs

Fractals

Looping Lemmas

Fractal Curves and Recursive Turtle Programs

Programming Projects

Some Strange Properties of Fractal Curves

Fractal Strangeness

Dimension

Differentiability

Attraction

Affine Transformations

Transformations

Conformal Transformations

Algebra of Affine Transformations

Geometry of Affine Transformations

Affine Coordinates and Affine Matrices

Conformal Transformations: Revisited

General Affine Transformations

Affine Geometry: A Connect-the-Dots Approach to Two-Dimensional Computer Graphics

Two Shortcomings of Turtle Graphics

Affine Graphics

Fractals from Iterated Function Systems

Generating Fractals by Iterating Transformations

Fractals as Fixed Points of Iterated Function Systems

Fractals as Attractors

Fractals with Condensation Sets

Programming Projects

Fixed-Point Theorem and Its Consequences

Fixed Points and Iteration

Trivial Fixed-Point Theorem

Consequences of the Trivial Fixed-Point Theorem

Programming Projects

Recursive Turtle Programs and Conformal Iterated Function Systems

Motivating Questions

Effect of Changing the Turtle’s Initial State

Equivalence Theorems

Conversion Algorithms

Bump Fractals

Programming Projects

Mathematical Methods for Three-Dimensional Computer Graphics

Vector Geometry: A Coordinate-Free Approach

Coordinate-Free Methods

Vectors and Vector Spaces

Points and Affine Spaces

Vector Products

Appendix A: The Nonassociativity of the Cross Product

Appendix B: The Algebra of Points and Vectors

Coordinate Algebra

Rectangular Coordinates

Addition, Subtraction, and Scalar Multiplication

Vector Products

Some Applications of Vector Geometry

Introduction

Trigonometric Laws

Representations for Lines and Planes

Metric Formulas

Intersection Formulas for Lines and Planes

Spherical Linear Interpolation

Inside-Outside Tests

Coordinate-Free Formulas for Affine and Projective Transformations

Transformations for Three-Dimensional Computer Graphics

Affine and Projective Transformations

Rigid Motions

Scaling

Projections

Matrix Representations for Affine and Projective Transformations

Matrix Representations for Affine Transformations

Linear Transformation Matrices and Translation Vectors

Rigid Motions

Scaling

Projections

Perspective

Programming Projects

Projective Space versus the Universal Space of Mass-Points

Algebra and Geometry

Projective Space: The Standard Model

Mass-Points: The Universal Model

Perspective and Pseudoperspective

Quaternions: Multiplication in the Space of Mass-Points

Vector Spaces and Division Algebras

Complex Numbers

Quaternions

Three-Dimensional Computer Graphics: Realistic Rendering

Color and Intensity

Introduction

RGB Color Model

Ambient Light

Diffuse Reflection

Specular Reflection

Total Intensity

Recursive Ray Tracing

Raster Graphics

Recursive Ray Tracing

Shadows

Reflection

Refraction

Surfaces I: The General Theory

Surface Representations

Surface Normals

Ray—Surface Intersections

Mean and Gaussian Curvature

Surfaces II: Simple Surfaces

Simple Surfaces

Intersection Strategies

Planes and Polygons

Natural Quadrics

General Quadric Surfaces

Tori

Surfaces of Revolution

Programming Projects

Solid Modeling

Solids

Constructive Solid Geometry

Boundary Representations

Octrees

Programming Projects

Shading

Polygonal Models

Uniform Shading

Gouraud Shading

Phong Shading

Programming Projects

Hidden Surface Algorithms

Hidden Surface Algorithms

The Heedless Painter

z-Buffer (Depth Buffer)

Scan Line

Ray Casting

Depth Sort

bsp-Tree

Programming Projects

Radiosity

Radiosity

Radiosity Equations

Form Factors

Radiosity Rendering Algorithm

Solving the Radiosity Equations

Programming Projects

Geometric Modeling: Freedom Curves and Surfaces

Bezier Curves and Surfaces

Interpolation and Approximation

de Casteljau Evaluation Algorithm

Bernstein Representation

Geometric Properties of Bezier Curves

Differentiating the de Casteljau Algorithm

Tensor Product Bezier Patches

Bezier Subdivision

Divide and Conquer

de Casteljau Subdivision Algorithm

Rendering and Intersection Algorithms

Variation Diminishing Property of Bezier Curves

Joining Bezier Curves Smoothly

Programming Projects

Blossoming

Motivation

Blossom

Blossoming and the de Casteljau Algorithm

Differentiation and the Homogeneous Blossom

B-Spline Curves and Surfaces

Motivation

Blossoming and the Local de Boor Algorithm

B-Spline Curves and the Global de Boor Algorithm

Smoothness

Labeling and Locality in the Global de Boor Algorithm

Every Spline Is a B-Spline

Geometric Properties of B-Spline Curves

Tensor Product B-Spline Surfaces

Nonuniform Rational B-Splines (NURBs)

Knot Insertion Algorithms for B-Spline Curves and Surfaces

Motivation

Knot Insertion

Local Knot Insertion Algorithms

Global Knot Insertion Algorithms

Programming Projects

Subdivision Matrices and Iterated Function Systems

Subdivision Algorithms and Fractal Procedures

Subdivision Matrices

Iterated Function Systems Built from Subdivision Matrices

Fractals with Control Points

Programming Projects

Subdivision Surfaces

Motivation

Box Splines

Quadrilateral Meshes

Triangular Meshes

Programming Projects

Further Readings

Index

A Summary and Exercises appear at the end of each chapter.