Mathematical Structures for Computer Graphics

A comprehensive exploration of the mathematics behind the modeling and rendering of computer graphics scenes Mathematical Structures for Computer Graphics presents an accessible and intuitive approach to the mathematical ideas and techniques necessary for two- and three-dimensional computer graphics. Focusing on the significant mathematical results, the book establishes key algorithms used to build complex graphics scenes. Written for readers with various levels of mathematical background, the book develops a solid foundation for graphics techniques and fills in relevant graphics details often overlooked in the literature. Rather than use a rigid theorem/proof approach, the book provides a flexible discussion that moves from vector geometry through transformations, curve modeling, visibility, and lighting models. Mathematical Structures for Computer Graphics also includes: Numerous examples of two- and three-dimensional techniques along with numerical calculations Plenty of mathematical and programming exercises in each chapter, which are designed particularly for graphics tasks Additional details at the end of each chapter covering historical notes, further calculations, and connected concepts for readers who wish to delve deeper Unique coverage of topics such as calculations with homogeneous coordinates, computational geometry for polygons, use of barycentric coordinates, various descriptions for curves, and L-system techniques for recursive images Mathematical Structures for Computer Graphics is an excellent textbook for undergraduate courses in computer science, mathematics, and engineering, as well as an ideal reference for practicing engineers, researchers, and professionals in computer graphics fields. The book is also useful for those readers who wish to understand algorithms for producing their own interesting computer images.

A comprehensive exploration of the mathematics behind the modeling and rendering of computer graphics scenesMathematical Structures for Computer Graphics presents an accessible and intuitive approach to the mathematical ideas and techniques necessary for two- and three-dimensional computer graphics. Focusing on the significant mathematical results, the book establishes key algorithms used to build complex graphics scenes.Written for readers with various levels of mathematical background, the book develops a solid foundation for graphics techniques and fills in relevant graphics details often overlooked in the literature. Rather than use a rigid theorem/proof approach, the book provides a flexible discussion that moves from vector geometry through transformations, curve modeling, visibility, and lighting models. Mathematical Structures for Computer Graphics also includes:* Numerous examples of two- and three-dimensional techniques along with numerical calculations* Plenty of mathematical and programming exercises in each chapter, which are designed particularly for graphics tasks* Additional details at the end of each chapter covering historical notes, further calculations, and connected concepts for readers who wish to delve deeper* Unique coverage of topics such as calculations with homogeneous coordinates, computational geometry for polygons, use of barycentric coordinates, various descriptions for curves, and L-system techniques for recursive imagesMathematical Structures for Computer Graphics is an excellent textbook for undergraduate courses in computer science, mathematics, and engineering, as well as an ideal reference for practicing engineers, researchers, and professionals in computer graphics fields. The book is also useful for those readers who wish to understand algorithms for producing their own interesting computer images.

Preface iii1 Basics 11.1 Graphics Pipeline 21.2 Mathematical Descriptions 51.3 Position 61.4 Distance 91.5 Complements and Details 131.6 Exercises 172 Vector Algebra 212.1 Basic Vector Characteristics 222.2 Two Important Products 312.3 Complements and Details 422.4 Exercises 463 Vector Geometry 493.1 Lines & Planes 493.2 Distances 553.3 Angles 633.4 Intersections 653.5 Additional Key Applications 733.6 Homogeneous Coordinates 863.7 Complements and Details 903.8 Exercises 944 Transformations 994.1 Types of Transformations 1004.2 Linear Transformations 1014.3 Three dimensions 1134.4 Affine Transformations 1234.5 Complements and Details 1344.6 Exercises 1455 Orientation 1495.1 Cartesian Coordinate Systems 1515.2 Cameras 1595.3 Other Coordinate Systems 1825.4 Complements and Details 1905.5 Exercises 1936 Polygons & Polyhedra 1976.1 Triangles 1976.2 Polygons 2136.3 Polyhedra 2306.4 Complements and Details 2456.5 Exercises 2507 Curves & Surfaces 2557.1 Curve Descriptions 2567.2 Bézier Curves 2687.3 B-Splines 2787.4 NURBS 2957.5 Surfaces 3007.6 Complements and Details 3117.7 Exercises 3168 Visibility 3218.1 Viewing 3218.2 Perspective Transformation 3238.3 Hidden Surfaces 3338.4 Ray Tracing 3448.5 Complements and Details 3518.6 Exercises 3569 Lighting 3599.1 Color Coordinates 3599.2 Elementary Lighting Models 3649.3 Global Illumination 3849.4 Textures 3919.5 Complements and Details 4039.6 Exercises 40810 Other Paradigms 41110.1 Pixels 41210.2 Noise 42110.3 L-Systems 43510.4 Exercises 443A Geometry & Trigonometry 447A.1 Triangles 447A.2 Angles 449A.3 Trigonometric Functions 450B Linear Algebra 455B.1 Systems of Linear Equations 455B.2 Matrix Properties 458B.3 Vector Spaces 460