Calculus for Computer Graphics

Students studying different branches of computer graphics need to be familiar with geometry, matrices, vectors, rotation transforms, quaternions, curves and surfaces. And as computer graphics software becomes increasingly sophisticated, calculus is also being used to resolve its associated problems. In this 3 rd edition, the author extends the scope of the original book to include vector differential operators and differential equations and draws upon his experience in teaching mathematics to undergraduates to make calculus appear no more challenging than any other branch of mathematics. He introduces the subject by examining how functions depend upon their independent variables, and then derives the appropriate mathematical underpinning and definitions. This gives rise to a function’s derivative and its antiderivative, or integral. Using the idea of limits, the reader is introduced to derivatives and integrals of many common functions. Other chapters address higher-order derivatives, partial derivatives, Jacobians, vector-based functions, single, double and triple integrals, with numerous worked examples and almost two hundred colour illustrations. This book complements the author’s other books on mathematics for computer graphics and assumes that the reader is familiar with everyday algebra, trigonometry, vectors and determinants. After studying this book, the reader should understand calculus and its application within the world of computer graphics, games and animation. Preface Contents 1 Introduction 1.1 What Is Calculus? 1.2 Where Is Calculus Used in Computer Graphics? 1.3 Who Else Should Read This Book? 1.4 Who Invented Calculus? 2 Functions 2.1 Introduction 2.2 Expressions, Variables, Constants and Equations 2.3 Functions 2.3.1 Continuous and Discontinuous Functions 2.3.2 Linear Graph Functions 2.3.3 Periodic Functions 2.3.4 Polynomial Functions 2.3.5 Function of a Function 2.3.6 Other Functions 2.4 A Function's Rate of Change 2.4.1 Slope of a Function 2.4.2 Differentiating Periodic Functions 2.5 Summary 3 Limits and Derivatives 3.1 Introduction 3.2 Some History of Calculus 3.3 Small Numerical Quantities 3.4 Equations and Limits 3.4.1 Quadratic Function 3.4.2 Cubic Equation 3.4.3 Functions and Limits 3.4.4 Graphical Interpretation of the Derivative 3.4.5 Derivatives and Differentials 3.4.6 Integration and Antiderivatives 3.5 Summary 3.6 Worked Examples 3.6.1 Limiting Value of a Quotient 1 3.6.2 Limiting Value of a Quotient 2 3.6.3 Derivative 3.6.4 Slope of a Polynomial 3.6.5 Slope of a Periodic Function 3.6.6 Integrate a Polynomial References 4 Derivatives and Antiderivatives 4.1 Introduction 4.2 Differentiating Groups of Functions 4.2.1 Sums of Functions 4.2.2 Function of a Function 4.2.3 Function Products 4.2.4 Function Quotients 4.2.5 Summary: Groups of Functions 4.3 Differentiating Implicit Functions 4.4 Differentiating Exponential and Logarithmic Functions 4.4.1 Exponential Functions 4.4.2 Logarithmic Functions 4.4.3 Summary: Exponential and Logarithmic Functions 4.5 Differentiating Trigonometric Functions 4.5.1 Differentiating tan 4.5.2 Differentiating csc 4.5.3 Differentiating sec 4.5.4 Differentiating cot 4.5.5 Differentiating arcsin, arccos and arctan 4.5.6 Differentiating arccsc, arcsec and arccot 4.5.7 Summary: Trigonometric Functions 4.6 Differentiating Hyperbolic Functions 4.6.1 Differentiating sinh, cosh and tanh 4.6.2 Differentiating cosech, sech and coth 4.6.3 Differentiating arsinh, arcosh and artanh 4.6.4 Differentiating arcsch, arsech and arcoth 4.6.5 Summary: Hyperbolic Functions 4.7 Summary 5 Higher Derivatives 5.1 Introduction 5.2 Higher Derivatives of a Polynomial 5.3 Identifying a Local Maximum or Minimum 5.4 Derivatives and Motion 5.5 Summary 5.5.1 Summary of Formulae 6 Partial Derivatives 6.1 Introduction 6.2 Partial Derivatives 6.2.1 Visualising Partial Derivatives 6.2.2 Mixed Partial Derivatives 6.3 Chain Rule 6.4 Total Derivative 6.5 Second-Order and Higher Partial Derivatives 6.6 Summary 6.6.1 Summary of Formulae 6.7 Worked Examples 6.7.1 Partial Derivative 6.7.2 First and Second-Order Partial Derivatives 6.7.3 Mixed Partial Derivative 6.7.4 Chained Partial Derivatives 6.7.5 Total Derivative 7 Integral Calculus 7.1 Introduction 7.2 Indefinite Integral 7.3 Standard Integration Formulae 7.4 Integrating Techniques 7.4.1 Continuous Functions 7.4.2 Difficult Functions 7.4.3 Trigonometric Identities 7.4.4 Exponent Notation 7.4.5 Completing the Square 7.4.6 The Integrand Contains a Derivative 7.4.7 Converting the Integrand into a Series of Fractions 7.4.8 Integration by Parts 7.4.9 Integrating by Substitution 7.4.10 Partial Fractions 7.5 Summary 7.6 Worked Examples 7.6.1 Trigonometric Identities 7.6.2 Exponent Notation 7.6.3 Completing the Square 7.6.4 The Integrand Contains a Derivative 7.6.5 Converting the Integrand into a Series of Fractions 7.6.6 Integration by Parts 7.6.7 Integrating by Substitution 7.6.8 Partial Fractions 8 Area Under a Graph 8.1 Introduction 8.2 Calculating Areas 8.3 Positive and Negative Areas 8.4 Area Between Two Functions 8.5 Areas with the y-Axis 8.6 Area with Parametric Functions 8.7 Bernhard Riemann 8.7.1 Domains and Intervals 8.7.2 The Riemann Sum 8.8 Summary Reference 9 Arc Length and Parameterisation of Curves 9.1 Introduction 9.2 Lagrange's Mean-Value Theorem 9.3 Arc Length 9.3.1 Arc Length of a Straight Line 9.3.2 Arc Length of a Circle 9.3.3 Arc Length of a Parabola 9.3.4 Arc Length of y=x32 9.3.5 Arc Length of a Sine Curve 9.3.6 Arc Length of a Hyperbolic Cosine Function 9.3.7 Arc Length of Parametric Functions 9.3.8 Arc Length of a Circle 9.3.9 Arc Length of an Ellipse 9.3.10 Arc Length of a Helix 9.3.11 Arc Length of a 2D Quadratic Bézier Curve 9.3.12 Arc Length of a 3D Quadratic Bézier Curve 9.3.13 Arc Length Parameterisation of a 3D Line 9.3.14 Arc Length Parameterisation of a Helix 9.3.15 Positioning Points on a Straight Line Using a Square Law 9.3.16 Positioning Points on a Helix Curve Using a Square Law 9.3.17 Arc Length Using Polar Coordinates 9.4 Summary 9.4.1 Summary of Formulae 9.5 Worked Examples 9.5.1 Arc Length of a Straight Line 9.5.2 Arc Length of a Circle 9.5.3 Arc Length of y=2x32 9.5.4 Arc Length of a Helix References 10 Surface Area 10.1 Introduction 10.2 Surface of Revolution 10.2.1 Surface Area of a Cylinder 10.2.2 Surface Area of a Right Cone 10.2.3 Surface Area of a Sphere 10.2.4 Surface Area of a Paraboloid 10.3 Surface Area Using Parametric Functions 10.4 Double Integrals 10.5 Jacobians 10.5.1 1D Jacobian 10.5.2 2D Jacobian 10.5.3 3D Jacobian 10.6 Double Integrals for Calculating Area 10.7 Summary 10.7.1 Summary of Formulae 10.8 Worked Examples 10.8.1 Surface Area of a Cylinder 10.8.2 Surface Area Swept Out by a Function 10.8.3 Double Integrals 10.8.4 Area Using Double Integral Reference 11 Volume 11.1 Introduction 11.2 Solid of Revolution: Disks 11.2.1 Volume of a Cylinder 11.2.2 Volume of a Right Cone 11.2.3 Volume of a Right Conical Frustum 11.2.4 Volume of a Sphere 11.2.5 Volume of an Ellipsoid 11.2.6 Volume of a Paraboloid 11.3 Solid of Revolution: Shells 11.3.1 Volume of a Cylinder 11.3.2 Volume of a Right Cone 11.3.3 Volume of a Hemisphere 11.3.4 Volume of a Paraboloid 11.4 Volumes with Double Integrals 11.4.1 Objects with a Rectangular Base 11.4.2 Rectangular Box 11.4.3 Rectangular Prism 11.4.4 Curved Top 11.4.5 Objects with a Circular Base 11.4.6 Cylinder 11.4.7 Truncated Cylinder 11.5 Volumes with Triple Integrals 11.5.1 Rectangular Box 11.5.2 Volume of a Cylinder 11.5.3 Volume of a Sphere 11.5.4 Volume of a Cone 11.6 Summary 11.6.1 Summary of Formulae 11.7 Worked Examples 11.7.1 Volume of a Cylinder 11.7.2 Volume of a Right Cone 11.7.3 Quadratic Rectangular Prism 11.7.4 Curved Top 11.7.5 Cylinder with a Curved Top 12 Vector-Valued Functions 12.1 Introduction 12.2 Differentiating Vector Functions 12.2.1 Velocity and Speed 12.2.2 Acceleration 12.2.3 Rules for Differentiating Vector-Valued Functions 12.3 Integrating Vector-Valued Functions 12.3.1 Distance Fallen by an Object 12.3.2 Position of a Moving Object 12.4 Summary 12.4.1 Summary of Formulae 12.5 Worked Examples 12.5.1 Differentiating a Position Vector 12.5.2 Speed of an Object at Different Times 12.5.3 Velocity and Acceleration of an Object at Different Times 12.5.4 Distance Fallen by an Object 12.5.5 Position of a Moving Object 13 Vector Differential Operators 13.1 Introduction 13.2 Scalar Fields 13.3 Vector Fields 13.4 The Gradient of a Scalar Field 13.4.1 Gradient of a Scalar Field in mathbbR2 13.4.2 Gradient of a Scalar Field in mathbbR3 13.4.3 Surface Normal Vectors 13.5 The Divergence of a Vector Field 13.6 Curl of a Vector Field 13.7 Summary 13.7.1 Summary of Formulae 13.8 Worked Examples 13.8.1 Gradient of a Scalar Field 13.8.2 Normal Vector to an Ellipse 13.8.3 Divergence of a Vector Field 13.8.4 Curl of a Vector Field 14 Tangent and Normal Vectors 14.1 Introduction 14.2 Notation 14.3 Tangent Vector to a Curve 14.4 Normal Vector to a Curve 14.4.1 Unit Tangent and Normal Vectors to a Line 14.4.2 Unit Tangent and Normal Vectors to a Parabola 14.4.3 Unit Tangent and Normal Vectors to a Circle 14.4.4 Unit Tangent and Normal Vectors to an Ellipse 14.4.5 Unit Tangent and Normal Vectors to a Sine Curve 14.4.6 Unit Tangent and Normal Vectors to a Cosh Curve 14.4.7 Unit Tangent and Normal Vectors to a Helix 14.4.8 Unit Tangent and Normal Vectors to a Quadratic Bézier Curve 14.5 Unit Tangent and Normal Vectors to a Surface 14.5.1 Unit Normal Vectors to a Bilinear Patch 14.5.2 Unit Normal Vectors to a Quadratic Bézier Patch 14.5.3 Unit Tangent and Normal Vector to a Sphere 14.5.4 Unit Tangent and Normal Vectors to a Torus 14.6 Summary 14.6.1 Summary of Formulae 15 Continuity 15.1 Introduction 15.2 B-Splines 15.2.1 Uniform B-Splines 15.2.2 B-Spline Continuity 15.3 Derivatives of a Bézier Curve 15.4 Summary 16 Curvature 16.1 Introduction 16.2 Curvature 16.2.1 Curvature of a Circle 16.2.2 Curvature of a Helix 16.2.3 Curvature of a Parabola 16.2.4 Parametric Plane Curve 16.2.5 Curvature of a Graph 16.2.6 Curvature of a 2D Quadratic Bézier Curve 16.2.7 Curvature of a 2D Cubic Bézier Curve 16.3 Summary 16.3.1 Summary of Formulae 16.4 Worked Examples 16.4.1 Curvature of a Circle 16.4.2 Curvature of a Helix 17 Solving Differential Equations 17.1 Introduction 17.2 What Is a Differential Equation? 17.3 Basic Concepts 17.3.1 Order and Degree 17.3.2 General Solution to a Differential Equation 17.4 Solving First-Order Ordinary Differential Equations 17.4.1 Separation of Variables 17.4.2 Substitution of Variables 17.4.3 Integrating Factor 17.5 Applications 17.5.1 Growth Models 17.5.2 Compound Interest 17.5.3 Radiocarbon Dating 17.6 Worked Examples 17.6.1 Direct Integration 17.6.2 Separation of Variables 1 17.6.3 Separation of Variables 2 17.6.4 Substitution of Variables 17.6.5 Integrating Factor 1 17.6.6 Integrating Factor 2 17.6.7 Compound Interest 17.6.8 Radiocarbon Dating References 18 Conclusion Appendix A Limit of (sinθ)/θ Appendix B Integrating cosnθ Index