Mathematical and Computer Programming Techniques for 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 MatrixA 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 Transformation