Mathematical & Computer Programming Techniques for Computer Graphics

Description

Mathematical and Computer Programming Techniques for Computer Graphics introduces the mathematics and related computer programming techniques used in Computer Graphics. Starting with the underlying mathematical ideas, it gradually leads the reader to a sufficient understanding of the detail to be able to implement libraries and programs for 2D and 3D graphics. Using lots of code examples, the reader is encouraged to explore and experiment with data and computer programs (in the C programming language) and to master the related mathematical techniques.

A simple but effective set of routines are included, organised as a library, covering both 2D and 3D graphics - taking a parallel approach to mathematical theory, and showing the reader how to incorporate it into example programs. This approach both demystifies the mathematics and demonstrates its relevance to 2D and 3D computer graphics.

Vector Algebra Survival Kit

Some Basic Definitions and Notation

Multiplication of a Vector by a Scalar

Vector Addition

Position Vectors and Free Vectors

The Vector Equation of a Line

Linear Dependence / Independence of Vectors

Vector Bases

The Components of a Vector

Multiplication of a Vector by a Scalar

Vector Addition

Vector Equality

Orthogonal, Orthonormal and Right-Handed Vector Bases

Cartesian Bases and Cartesian Coordinates

The Length of a Vector

The Scalar Product of Vectors

The Scalar Product Expresses in Terms of its Components

Properties and Applications of the Scalar Product

The Direction Ratios and Direction Cosines of a Vector

The Vector Product of Two Vectors

The Vector Product Expressed in Terms of its Components

Properties of the Vector Product

Triple Produces of Vectors

The Components of a Vector Relative to a Non-orthogonal Basis

The Decomposition of a Vector According to a Basis

The Vector Equation of the Line Revisited

The Vector Equation of the Place

Some Applications of Vector Algebra in Analytical Geometry

Summary of Vector Algebra Axioms and Rules

A Simple Vector Algebra C Library

Matrix Algebra Survival Kit

The Definition of a Matrix

Square Matrices

Diagonal Matrices

The Identity Matrix

The Zero or Null Matrix

The Transpose of a Matrix

Symmetric and Antisymmetric Matrices

Triangular Matrices

Scalar Matrices

Equality of Matrices

Matrix Operations

The Minor of a Matrix

The Determinant of a Matrix

The Computational Rules of Determinants

The Cofactor of an Element of a Matrix and the Cofactor Matrix

The Ajoint Matrix or Adjugate Matrix

The Reciprocal or Inverse of a Matrix

A Theorem on Invertible Matrices and their Determinants

Axioms and Rules of Matrix Inversion

Solving a System of Linear Simultaneous Equations

Orthogonal Matrices

Two Theorems on Vector by Matrix Multiplication

The Row / Column Reversal Matrix

Summary of Matrix Algebra Axioms and Rules

A Simple Matrix Algebra C Library

Vector Spaces or Linear Spaces

The Definition of a Scalar Field

The Definition of a Vector Space

Linear Combinations of Vectors

Linear Dependence and Linear Independence of Vectors

Spans and Bases of a Vector Space

Transformations between Bases

Transformations between Orthonormal Bases

An Alternative Notation for Change of Basis Transformations

Two-Dimensional Transformations

The Definition of a 2D Transformation

The Concatenation of Transformations

2D Graphics Transformations

2D Primitive Transformations

2D Composite Transformations

The Sign of the Angles in Transformations

Some Important Observations

The Matrix Representation of 2D Transformations

The Matrix Representation of Primitive Transformations

Some Transformation Matrix Properties

The Concatenation of Transformation Matrices

Local Frame and Global Frame Transformations

Transformations of the Frame of Reference or Coordinate System

The Viewing Transformation

Homogeneous Coordinates

A Simple C Library for 2D Transformations

Two-Dimensional Clipping

Clipping a 2D Point to a Rectangular Clipping Boundary

Clipping a 2D Line Segment to a Rectangular Clipping Boundary

The Cohen and Sutherland 2D Line-Clipping Algorithm

2D Polygon Clipping

References

Three-Dimensional Transformations

Primitive 3D Transformations

The Global and Local Frames of Reference

Aiming Transformations

Composite Transformations

Local Frame and Global Frame Transformations

Transformations of the Frame of Reference or Coordinate System

References

Viewing and Projection Transformations

The Conceptual Camera Model

The Viewing Transformation

The Projection Transformation

The Projection Transformation Matrix

Parallel Projections

Perspective Projections

The Screen or Device Coordinate System

3D Line Clipping

Perspective Depth

A Simple C Library for 3D Transformations

3D Rendering

Introduction

Rendering Algorithms

Reflection Models and Shading Techniques

Shading Models

References

A1: A Simple Vector Algebra C Library

A2: A Simple Matrix Algebra C Library

A3: A Simple C Library for 2D Transformations

A4: A Simple C Library for 3D Transformati