Calculus for Computer Graphics

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