Vectors and Matrices for Geometric and 3D Modeling

by Michael Mortenson

Overview

Vectors are perhaps the most important mathematical objects used in modeling and animation. They have the properties of magnitude and direction, and provide visual understanding of model construction and analysis. Matrices are natural and hardworking partners of vectors.

This work presents lessons on vectors and matrices in geometric and 3D modeling—the mathematics at the foundation of computer graphics applications. The lessons appear as chapters, generally organized from introductory to more complex topics. Within each chapter, there is a similar order of elementary-to-advanced discussion. 

Here are topics that are usually published in briefer form in more advanced texts as part of their supporting mathematics. In this work, vectors and matrices are the main subjects. This text offers an easier-to-understand introduction to the main ideas behind vectors and matrices, stripped of formalism, and leading directly to geometric modeling. It demonstrates the relationships between vectors, matrices, basis vectors and barycentric coordinates, all of which are not usually seen in ordinary texts. This text also discusses how these concepts are applied to produce curves and surfaces, and how they facilitate the analysis of spatial relationships. 

For those readers beginning studies in geometric and 3D modeling, animation, CGI, or CAD/CAM, this work serves as an introduction to vectors and matrices, and provides a good start to understanding how they are applied. For instructors, this book can be a primary text or supplement to more advanced or specialized texts on geometric and 3D modeling.
           
Features

• More than 150 illustrations support visual understanding of the content.
• 100+ exercises and extended solutions enhance the classroom environment.
• A comprehensive range of topics offers an in-depth look at the math underlying 3D modeling and animation courses.
• Linear algebra and calculus are not prerequisites.

Product Details

ISBN-13:	9780831136550
Publisher:	Industrial Press, Inc.
Publication date:	10/29/2020
Edition description:	First Edition
Pages:	272
Product dimensions:	8.50(w) x 11.00(h) x 0.60(d)
Age Range:	18 Years

About the Author

Michael E. Mortenson does independent research and writes on topics in geometric and 3D modeling. He is a former research scientist with a major aerospace corporation, and the author of several successful textbooks, including Geometric Modeling, Third Edition, Mathematics for Computer Graphics Applications, 2e, Geometric Transformations for 3D Modeling, 2e, and 3D Modeling, Animation, and Rendering: An Illustrated Lexicon. These works have garnered over 2400 citations. He is a graduate of the UCLA School of Engineering.

Table of Contents

Acknowledgments xi

Preface xiii

1 Vectors PDQ 1

1.1 A Very Brief Overview 1

1.2 With Comments Added 2

1.3 And What Follows 4

1.4 Backstory: The Very Short Version 5

1.5 Notation…Motivation 6

Vectors 6

Matrices 8

Indexing 9

Curves and Surfaces 9

2 The Visual Vector 11

2.1 Directed Line Segment 11

2.2 Free and Fixed Vectors 14

2.3 Vector Addition 15

2.4 Vector Components 17

2.5 Vector Multiplication 18

2.6 A Geometry Problem 21

2.7 The Idea of a Vector as a Displacement 22

3 The Algebraic Vector 23

3.1 There's a Vector in Your Coordinate System 23

3.2 Form and Equality 25

3.3 Magnitude and Direction 26

Magnitude 27

Direction 28

Unit Vectors 29

3.4 Addition and Subtraction 30

3.5 Products of Vectors 31

Scalar Product 32

Vector Product 33

Scalar Triple Product 35

Vector Triple Product 35

3.6 Functions and Derivatives 36

3.7 Linear Vector Spaces 37

3.8 Vector Equations 39

3.9 Quaternions 40

3.10 Geometry Problems 44

Problem 1 The Law of Sines 44

Problem 2 The Area of a Triangle 45

Problem 3 A Property of Triangles 45

3.11 Exercises 46

4 Matrix Basics 51

4.1 Equality and Order 51

4.2 Row and Column Matrices 52

4.3 Transpose of a Matrix 53

4.4 Addition and Subtraction 54

4.5 Multiplication 55

4.6 Partitioning a Matrix 58

4.7 Summary of Matrix Properties 60

4.8 Exercises 60

5 Special Matrices 63

5.1 Identity Matrix 63

5.2 Diagonal Matrix 64

5.3 Trace of a Matrix 65

5.4 Symmetric Matrix 66

5.5 Orthogonal Matrix 66

5.6 Inverse of a Matrix 67

5.7 Collinear Vectors, Eigenvalues, and Eigenvectors 68

5.8 Similar Matrices 71

5.9 Determinants 72

5.10 Summary of Determinant Properties 74

5.11 Exercises 75

6 Basis Vectors 79

6.1 Oblique Coordinate Systems 79

6.2 Basis Vector Basics 80

6.3 Change of Basis 82

6.4 Reciprocal Basis Vectors 84

6.5 Orthogonal Basis Vectors and Matrices 86

6.6 Frames 86

7 Barycentric Coordinate Systems 89

7.1 Origins 89

7.2 Barycentric Coordinates 89

7.3 Relative to Two Points 90

7.4 Relative to a Triangle 92

7.5 Relative to a Tetrahedron 96

7.6 Generalized Barycentric Coordinates 97

8 Translation and Rotation 99

8.1 Translation 99

8.2 Rotation in the Plane 104

8.3 Rotation in Space 108

8.4 Rotation about an Arbitrary Axis 110

8.5 Eigenvectors and Equivalent Rotations 114

8.6 Rotation and Quaternions 115

8.7 Combining Translation and Rotation 116

8.8 Kinematics 121

8.9 Exercises 123

9 More Transformations 125

9.1 Translation 125

9.2 Rotation 126

9.3 Scaling 127

9.4 Shear 133

9.5 Reflection 135

9.6 Inversion 139

9.7 Projection 142

9.8 Sweep Transformations 148

Translational Sweep 150

Generalized Translational Sweep 150

Rotational Sweep 153

Translation with Deformation Sweep 154

9.9 Exercises 155

10 Vector-Defined Geometric Objects I 157

10.1 Points 157

10.2 Lines 158

Line Through a Point and Parallel to a Vector 158

Line Defined by Two Points 161

10.3 Planes 162

Through a Point and Parallel to Two Independent Vectors 162

Three-Point Definition 163

Point on the Plane and the Unit Normal to It 163

Normal Vector from the Origin to the Plane 164

10.4 Polygons 164

Definition of a Polygon 164

Properties of Polygons 165

Convex Hull of a Polygon 167

Triangles: A Very Special Polygon 168

Data Structure for Polygons 168

10.5 Polyhedra 169

Definition of a Polyhedron 169

Properties of Polyhedra 170

Convex Hull of a Polyhedron 171

Connectivity Matrix 171

Data Structure for Polyhedra 173

Euler's Formula for Simple Polyhedra 174

Euler's Formula for Nonsimple Polyhedra 174

10.6 Exercises 176

11 Vector-Defined Geometric Objects II 179

11.1 Curve-Defining Functions 179

11.2 Bézier Curves 182

11.3 Local Properties of a Curve 184

11.4 Global Properties of a Curve 193

11.5 Surface-Defining Functions 195

11.6 Local Properties of a Surface 196

11.7 Global Properties of a Surface 198

12 Spatial Relationships 201

12.1 Distance 201

12.2 Minimum Distances 202

Minimum Distance Between a Point and a Curve 203

Minimum Distance Between a Point and a Plane 203

Minimum Distance Between a Point and a Surface 204

Minimum Distance Between Two Curves 205

Minimum Distance Between a Curve and a Plane 205

Minimum Distance Between a Curve and a Surface 206

Minimum Distance Between Two Surfaces 207

12.3 Line Intersections 208

Intersection of Two Lines 208

Intersection of a Line and a Plane 209

Intersection of a Line and a Curve 210

Intersection of a Line and a Surface 211

12.4 Plane Intersections 212

Intersection of Two Planes 212

Intersection of a Plane and a Curve 213

Intersection of a Plane and a Surface 214

12.5 Curve Intersections 214

Intersection of Two Curves 214

Intersection of a Curve and a Surface 215

Solutions 217

Chapter 3 The Algebraic Vector 217

Chapter 4 Matrix Basics 224

Chapter 5 Special Matrices 227

Chapter 8 Translation and Rotation 233

Chapter 9 More Transformations 237

Chapter 10 Vector-Defined Geometric Objects I 240

Index 247