Mathematics for Computer Graphics

Description

John Vince explains a comprehensive range of mathematical techniques and problem-solving strategies associated with computer games, computer animation, special effects, virtual reality, CAD and other areas of computer graphics in this completely revised and expanded sixth edition. The first five chapters cover a general introduction, number sets, algebra, trigonometry and coordinate systems, which are employed in the following chapters on determinants, vectors, matrix algebra, complex numbers, geometric transforms, quaternion algebra, quaternions in space, interpolation, curves and patches, analytical geometry and barycentric coordinates. Following this, the reader is introduced to the relatively new subject of geometric algebra, followed by two chapters that introduce differential and integral calculus. Finally, there is a chapter on worked examples. Mathematics for Computer Graphics covers all of the key areas of the subject, including: • Number sets • Algebra • Trigonometry • Complex numbers • Coordinate systems • Determinants • Vectors • Quaternions • Matrix algebra • Geometric transforms • Interpolation • Curves and surfaces • Analytic geometry • Barycentric coordinates • Geometric algebra • Differential calculus • Integral calculus This sixth edition contains approximately 150 worked examples and over 330 colour illustrations, which are central to the author’s descriptive writing style. Mathematics for Computer Graphics provides a sound understanding of the mathematics required for computer graphics software and setting the scene for further reading of more advanced books and technical research papers Preface Contents 1 Introduction 1.1 Mathematics for Computer Graphics 1.2 Understanding Mathematics 1.3 What Makes Mathematics Difficult? 1.4 Background to This Book 1.5 How to Use This Book 1.6 Symbols and Notation References 2 Numbers 2.1 Introduction 2.2 Background 2.3 Counting 2.4 Sets of Numbers 2.5 Zero 2.6 Negative Numbers 2.6.1 The Arithmetic of Positive and Negative Numbers 2.7 Observations and Axioms 2.7.1 Commutative Law 2.7.2 Associative Law 2.7.3 Distributive Law 2.8 The Base of a Number System 2.8.1 Background 2.8.2 Octal Numbers 2.8.3 Binary Numbers 2.8.4 Hexadecimal Numbers 2.8.5 Adding Binary Numbers 2.8.6 Subtracting Binary Numbers 2.9 Types of Numbers 2.9.1 Natural Numbers 2.9.2 Integers 2.9.3 Rational Numbers 2.9.4 Irrational Numbers 2.9.5 Real Numbers 2.9.6 Algebraic and Transcendental Numbers 2.9.7 Imaginary Numbers 2.9.8 Complex Numbers 2.9.9 Transcendental and Algebraic Numbers 2.9.10 Infinity 2.10 Summary 2.11 Worked Examples 2.11.1 Algebraic Expansion 2.11.2 Binary Subtraction 2.11.3 Complex Numbers 2.11.4 Complex Rotation 3 Algebra 3.1 Introduction 3.2 Background 3.2.1 Solving the Roots of a Quadratic Equation 3.3 Indices 3.3.1 Laws of Indices 3.4 Logarithms 3.5 Further Notation 3.6 Functions 3.6.1 Explicit and Implicit Equations 3.6.2 Function Notation 3.6.3 Intervals 3.6.4 Function Domains and Ranges 3.6.5 Odd and Even Functions 3.6.6 Power Functions 3.7 Summary 3.8 Worked Examples 3.8.1 Algebraic Manipulation 3.8.2 Solving a Quadratic Equation 3.8.3 Factorising References 4 Trigonometry 4.1 Introduction 4.2 Background 4.3 Units of Angular Measurement 4.4 The Trigonometric Ratios 4.4.1 Domains and Ranges 4.5 Inverse Trigonometric Ratios 4.6 Trigonometric Identities 4.7 The Sine Rule 4.8 The Cosine Rule 4.9 Compound-Angle Identities 4.9.1 Double-Angle Identities 4.9.2 Multiple-Angle Identities 4.9.3 Half-Angle Identities 4.10 Perimeter Relationships 4.11 Summary References 5 Coordinate Systems 5.1 Introduction 5.2 Background 5.3 The Cartesian Plane 5.4 Function Graphs 5.5 Shape Representation 5.5.1 2D Polygons 5.5.2 Area of a Shape 5.6 Theorem of Pythagoras in 2D 5.7 3D Cartesian Coordinates 5.7.1 Theorem of Pythagoras in 3D 5.8 Polar Coordinates 5.9 Spherical Polar Coordinates 5.10 Cylindrical Coordinates 5.11 Summary 5.12 Worked Examples 5.12.1 Area of a Shape 5.12.2 Distance Between Two Points 5.12.3 Polar Coordinates 5.12.4 Spherical Polar Coordinates 5.12.5 Cylindrical Coordinates Reference 6 Determinants 6.1 Introduction 6.2 Linear Equations with Two Variables 6.3 Linear Equations with Three Variables 6.3.1 Sarrus's Rule 6.4 Mathematical Notation 6.4.1 Matrix 6.4.2 Order of a Determinant 6.4.3 Value of a Determinant 6.4.4 Properties of Determinants 6.5 Summary 6.6 Worked Examples 6.6.1 Determinant Expansion 6.6.2 Complex Determinant 6.6.3 Simple Expansion 6.6.4 Simultaneous Equations 7 Vectors 7.1 Introduction 7.2 Background 7.3 2D Vectors 7.3.1 Vector Notation 7.3.2 Graphical Representation of Vectors 7.3.3 Magnitude of a Vector 7.4 3D Vectors 7.4.1 Vector Manipulation 7.4.2 Scaling a Vector 7.4.3 Vector Addition and Subtraction 7.4.4 Position Vectors 7.4.5 Unit Vectors 7.4.6 Cartesian Vectors 7.4.7 Products 7.4.8 Scalar Product 7.4.9 The Dot Product in Lighting Calculations 7.4.10 The Scalar Product in Back-Face Detection 7.4.11 The Vector Product 7.4.12 The Right-Hand Rule 7.5 Deriving a Unit Normal Vector for a Triangle 7.6 Surface Areas 7.6.1 Calculating 2D Areas 7.7 Summary 7.8 Worked Examples 7.8.1 Position Vector 7.8.2 Unit Vector 7.8.3 Vector Magnitude 7.8.4 Angle Between Two Vectors 7.8.5 Vector Product References 8 Matrix Algebra 8.1 Introduction 8.2 Background 8.3 Matrix Notation 8.3.1 Matrix Dimension or Order 8.3.2 Square Matrix 8.3.3 Column Vector 8.3.4 Row Vector 8.3.5 Null Matrix 8.3.6 Unit Matrix 8.3.7 Trace 8.3.8 Determinant of a Matrix 8.3.9 Transpose 8.3.10 Symmetric Matrix 8.3.11 Antisymmetric Matrix 8.4 Matrix Addition and Subtraction 8.4.1 Scalar Multiplication 8.5 Matrix Products 8.5.1 Row and Column Vectors 8.5.2 Row Vector and a Matrix 8.5.3 Matrix and a Column Vector 8.5.4 Square Matrices 8.5.5 Rectangular Matrices 8.6 Inverse Matrix 8.6.1 Inverting a Pair of Matrices 8.7 Orthogonal Matrix 8.8 Diagonal Matrix 8.9 Summary 8.10 Worked Examples 8.10.1 Matrix Inversion 8.10.2 Identity Matrix 8.10.3 Solving Two Equations Using Matrices 8.10.4 Solving Three Equations Using Matrices 8.10.5 Solving Two Complex Equations 8.10.6 Solving Three Complex Equations 8.10.7 Solving Two Complex Equations 8.10.8 Solving Three Complex Equations 9 Complex Numbers 9.1 Introduction 9.2 Definition of a Complex Number 9.2.1 Addition and Subtraction of Complex Numbers 9.2.2 Multiplying a Complex Number by a Scalar 9.2.3 Product of Complex Numbers 9.2.4 Square of a Complex Number 9.2.5 Norm of a Complex Number 9.2.6 Complex Conjugate of a Complex Number 9.2.7 Quotient of Complex Numbers 9.2.8 Inverse of a Complex Number 9.2.9 Square-Root of pmi 9.3 Ordered Pairs 9.3.1 Addition and Subtraction of Ordered Pairs 9.3.2 Multiplying an Ordered Pair by a Scalar 9.3.3 Product of Ordered Pairs 9.3.4 Square of an Ordered Pair 9.3.5 Norm of an Ordered Pair 9.3.6 Complex Conjugate of an Ordered Pair 9.3.7 Quotient of an Ordered Pair 9.3.8 Inverse of an Ordered Pair 9.3.9 Square-Root of pmi 9.4 Matrix Representation of a Complex Number 9.4.1 Adding and Subtracting Complex Numbers 9.4.2 Product of Two Complex Numbers 9.4.3 Norm Squared of a Complex Number 9.4.4 Complex Conjugate of a Complex Number 9.4.5 Inverse of a Complex Number 9.4.6 Quotient of a Complex Number 9.4.7 Square-Root of pmi 9.5 Summary 9.6 Worked Examples 9.6.1 Adding and Subtracting Complex Numbers 9.6.2 Product of Complex Numbers 9.6.3 Multiplying a Complex Number by i 9.6.4 The Norm of a Complex Number 9.6.5 The Complex Conjugate of a Complex Number 9.6.6 The Quotient of Two Complex Numbers 9.6.7 Divide a Complex Number by i 9.6.8 Divide a Complex Number by -i 9.6.9 The Inverse of a Complex Number 9.6.10 The Inverse of i 9.6.11 The Inverse of -i Reference 10 Geometric Transforms 10.1 Introduction 10.2 Background 10.3 2D Transforms 10.3.1 Translation 10.3.2 Scaling 10.3.3 Reflection 10.4 Transforms as Matrices 10.4.1 Systems of Notation 10.5 Homogeneous Coordinates 10.5.1 2D Translation 10.5.2 2D Scaling 10.5.3 2D Reflections 10.5.4 2D Shearing 10.5.5 2D Rotation 10.5.6 2D Scaling 10.5.7 2D Reflection 10.5.8 2D Rotation About an Arbitrary Point 10.6 3D Transforms 10.6.1 3D Translation 10.6.2 3D Scaling 10.6.3 3D Rotation 10.6.4 Gimbal Lock 10.6.5 Rotating About an Axis 10.6.6 3D Reflections 10.7 Change of Axes 10.7.1 2D Change of Axes 10.7.2 Direction Cosines 10.7.3 3D Change of Axes 10.8 Positioning the Virtual Camera 10.8.1 Direction Cosines 10.8.2 Euler Angles 10.9 Rotating a Point About an Arbitrary Axis 10.9.1 Matrices 10.10 Transforming Vectors 10.11 Determinants 10.12 Perspective Projection 10.13 Summary 10.14 Worked Examples 10.14.1 2D Scaling Transform 10.14.2 2D Scale and Translate 10.14.3 3D Scaling Transform 10.14.4 2D Rotation 10.14.5 2D Rotation About a Point 10.14.6 Determinant of the Rotate Transform 10.14.7 Determinant of the Shear Transform 10.14.8 Yaw, Pitch and Roll Transforms 10.14.9 3D Rotation About an Axis 10.14.10 3D Rotation Transform Matrix 10.14.11 2D Change of Axes 10.14.12 3D Change of Axes 10.14.13 Rotate a Point About an Axis 10.14.14 Perspective Projection 11 Quaternion Algebra 11.1 Introduction 11.2 Some History 11.3 Defining a Quaternion 11.3.1 The Quaternion Units 11.3.2 Example of Quaternion Products 11.4 Algebraic Definition 11.5 Adding and Subtracting Quaternions 11.6 Real Quaternion 11.7 Multiplying a Quaternion by a Scalar 11.8 Pure Quaternion 11.9 Unit Quaternion 11.10 Additive Form of a Quaternion 11.11 Binary Form of a Quaternion 11.12 The Complex Conjugate of a Quaternion 11.13 Norm of a Quaternion 11.14 Normalised Quaternion 11.15 Quaternion Products 11.15.1 Product of Pure Quaternions 11.15.2 Product of Unit-Norm Quaternions 11.15.3 Square of a Quaternion 11.15.4 Norm of the Quaternion Product 11.16 Inverse Quaternion 11.17 Matrices 11.17.1 Orthogonal Matrix 11.18 Quaternion Algebra 11.19 Summary 11.19.1 Summary of Definitions 11.20 Worked Examples 11.20.1 Adding and Subtracting Quaternions 11.20.2 Norm of a Quaternion 11.20.3 Unit-norm Quaternions 11.20.4 Quaternion Product 11.20.5 Square of a Quaternion 11.20.6 Inverse of a Quaternion References 12 Quaternions in Space 12.1 Introduction 12.2 Some History 12.3 Quaternion Products 12.3.1 Special Case 12.3.2 General Case 12.3.3 Double Angle 12.4 Quaternions in Matrix Form 12.4.1 Vector Method 12.4.2 Geometric Verification 12.5 Multiple Rotations 12.6 Rotating About an Off-Set Axis 12.7 Converting a Rotation Matrix to a Quaternion 12.8 Summary 12.8.1 Summary of Definitions 12.9 Worked Examples 12.9.1 Special Case Quaternion 12.9.2 Rotating a Vector Using a Quaternion 12.9.3 Evaluate qpq-1 12.9.4 Evaluate qpq-1 Using a Matrix References 13 Interpolation 13.1 Introduction 13.2 Background 13.3 Linear Interpolation 13.4 Non-Linear Interpolation 13.4.1 Trigonometric Interpolation 13.4.2 Cubic Interpolation 13.5 Interpolating Vectors 13.6 Interpolating Quaternions 13.7 Summary 14 Curves and Patches 14.1 Introduction 14.2 Background 14.3 The Circle 14.4 The Ellipse 14.5 Bézier Curves 14.5.1 Bernstein Polynomials 14.5.2 Quadratic Bézier Curves 14.5.3 Cubic Bernstein Polynomials 14.6 A Recursive Bézier Formula 14.7 Bézier Curves Using Matrices 14.7.1 Linear Interpolation 14.8 B-Splines 14.8.1 Uniform B-Splines 14.8.2 Continuity 14.8.3 Non-uniform B-Splines 14.8.4 Non-uniform Rational B-Splines 14.9 Surface Patches 14.9.1 Planar Surface Patch 14.9.2 Quadratic Bézier Surface Patch 14.9.3 Cubic Bézier Surface Patch 14.10 Summary 15 Analytic Geometry 15.1 Introduction 15.2 Background 15.2.1 Angles 15.2.2 Intercept Theorems 15.2.3 Golden Section 15.2.4 Triangles 15.2.5 Centre of Gravity of a Triangle 15.2.6 Isosceles Triangle 15.2.7 Equilateral Triangle 15.2.8 Right Triangle 15.2.9 Theorem of Thales 15.2.10 Theorem of Pythagoras 15.2.11 Quadrilateral 15.2.12 Trapezoid 15.2.13 Parallelogram 15.2.14 Rhombus 15.2.15 Regular Polygon 15.2.16 Circle 15.3 2D Analytic Geometry 15.3.1 Equation of a Straight Line 15.3.2 The Hessian Normal Form 15.3.3 Space Partitioning 15.3.4 The Hessian Normal Form from Two Points 15.4 Intersection Points 15.4.1 Intersecting Straight Lines 15.4.2 Intersecting Line Segments 15.5 Point Inside a Triangle 15.5.1 Area of a Triangle 15.5.2 Hessian Normal Form 15.6 Intersection of a Circle with a Straight Line 15.7 3D Geometry 15.7.1 Equation of a Straight Line 15.7.2 Intersecting Two Straight Lines 15.8 Equation of a Plane 15.8.1 Cartesian Form of the Plane Equation 15.8.2 General Form of the Plane Equation 15.8.3 Parametric Form of the Plane Equation 15.8.4 Converting from the Parametric to the General Form 15.8.5 Plane Equation from Three Points 15.9 Intersecting Planes 15.9.1 Intersection of Three Planes 15.9.2 Angle Between Two Planes 15.9.3 Angle Between a Line and a Plane 15.9.4 Intersection of a Line with a Plane 15.10 Summary 16 Barycentric Coordinates 16.1 Introduction 16.2 Background 16.3 Ceva's Theorem 16.4 Ratios and Proportion 16.5 Mass Points 16.6 Linear Interpolation 16.7 Convex Hull Property 16.8 Areas 16.9 Volumes 16.10 Bézier Curves and Patches 16.11 Summary Reference 17 Geometric Algebra 17.1 Introduction 17.2 Background 17.3 Symmetric and Antisymmetric Functions 17.4 Trigonometric Foundations 17.5 Vectorial Foundations 17.6 Inner and Outer Products 17.7 The Geometric Product in 2D 17.8 The Geometric Product in 3D 17.9 The Outer Product of Three 3D Vectors 17.10 Axioms 17.11 Notation 17.12 Grades, Pseudoscalars and Multivectors 17.13 Redefining the Inner and Outer Products 17.14 The Inverse of a Vector 17.15 The Imaginary Properties of the Outer Product 17.16 Duality 17.17 The Relationship Between the Vector Product and the Outer Product 17.18 The Relationship Between Quaternions and Bivectors 17.19 Reflections and Rotations 17.19.1 2D Reflections 17.19.2 3D Reflections 17.19.3 2D Rotations 17.20 Rotors 17.21 Applied Geometric Algebra 17.22 Summary References 18 Calculus: Derivatives 18.1 Introduction 18.2 Background 18.3 Small Numerical Quantities 18.4 Equations and Limits 18.4.1 Quadratic Function 18.4.2 Cubic Equation 18.4.3 Functions and Limits 18.4.4 Graphical Interpretation of the Derivative 18.4.5 Derivatives and Differentials 18.4.6 Integration and Antiderivatives 18.5 Function Types 18.6 Differentiating Groups of Functions 18.6.1 Sums of Functions 18.6.2 Function of a Function 18.6.3 Function Products 18.6.4 Function Quotients 18.7 Differentiating Implicit Functions 18.8 Differentiating Exponential and Logarithmic Functions 18.8.1 Exponential Functions 18.8.2 Logarithmic Functions 18.9 Differentiating Trigonometric Functions 18.9.1 Differentiating tan 18.9.2 Differentiating csc 18.9.3 Differentiating sec 18.9.4 Differentiating cot 18.9.5 Differentiating arcsin, arccos and arctan 18.9.6 Differentiating arccsc, arcsec and arccot 18.10 Differentiating Hyperbolic Functions 18.10.1 Differentiating sinh, cosh and tanh 18.11 Higher Derivatives 18.12 Higher Derivatives of a Polynomial 18.13 Identifying a Local Maximum or Minimum 18.14 Partial Derivatives 18.14.1 Visualising Partial Derivatives 18.14.2 Mixed Partial Derivatives 18.15 Chain Rule 18.16 Total Derivative 18.17 Summary Reference 19 Calculus: Integration 19.1 Introduction 19.2 Indefinite Integral 19.3 Integration Techniques 19.3.1 Continuous Functions 19.3.2 Difficult Functions 19.3.3 Trigonometric Identities 19.3.4 Exponent Notation 19.3.5 Completing the Square 19.3.6 The Integrand Contains a Derivative 19.3.7 Converting the Integrand into a Series of Fractions 19.3.8 Integration by Parts 19.3.9 Integration by Substitution 19.3.10 Partial Fractions 19.4 Area Under a Graph 19.5 Calculating Areas 19.6 Positive and Negative Areas 19.7 Area Between Two Functions 19.8 Areas with the y-Axis 19.9 Area with Parametric Functions 19.10 The Riemann Sum 19.11 Summary 20 Worked Examples 20.1 Introduction 20.2 Area of Regular Polygon 20.3 Area of Any Polygon 20.4 Dihedral Angle of a Dodecahedron 20.5 Vector Normal to a Triangle 20.6 Area of a Triangle Using Vectors 20.7 General Form of the Line Equation from Two Points 20.8 Angle Between Two Straight Lines 20.9 Test if Three Points Lie on a Straight Line 20.10 Position and Distance of the Nearest Point on a Line to a Point 20.11 Position of a Point Reflected in a Line 20.12 Intersection of a Line and a Sphere 20.13 Sphere Touching a Plane 20.14 Summary Appendix A Limit of (sinθ)/θ Appendix B Integrating cosnθ Index