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Calculus for Computer Graphics

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Students studying different branches of computer graphics have 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 2nd edition, the author extends the scope of the original book to include applications of calculus in the areas of arc-length parameterisation of curves, geometric continuity, tangent and normal vectors, and curvature. The author 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 over a hundred and seventy 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.

Author(s): John Vince
Edition: 2
Publisher: Springer
Year: 2019

Language: English
Pages: 320
Tags: computer, graphics

Preface
Contents
1 Introduction
1.1 What is Calculus?
1.2 Where is Calculus Used in Computer Graphics?
1.3 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 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 Small Numerical Quantities
3.3 Equations and Limits
3.3.1 Quadratic Function
3.3.2 Cubic Equation
3.3.3 Functions and Limits
3.3.4 Graphical Interpretation of the Derivative
3.3.5 Derivatives and Differentials
3.3.6 Integration and Antiderivatives
3.4 Summary
3.5 Worked Examples
3.5.1 Limiting Value of a Quotient
3.5.2 Limiting Value of a Quotient
3.5.3 Derivative
3.5.4 Slope of a Polynomial
3.5.5 Slope of a Periodic Function
3.5.6 Integrate a Polynomial
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
7 Integral Calculus
7.1 Introduction
7.2 Indefinite Integral
7.3 Standard Integration Formulae
7.4 Integration 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 Integration by Substitution
7.4.10 Partial Fractions
7.5 Summary
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
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
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
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 Sphere
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
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 Velocity of a Falling Object
12.3.2 Position of a Moving Object
12.4 Summary
12.4.1 Summary of Formulae
13 Tangent and Normal Vectors
13.1 Introduction
13.2 Notation
13.3 Tangent Vector to a Curve
13.4 Normal Vector to a Curve
13.5 Gradient of a Scalar Field
13.5.1 Unit Tangent and Normal Vectors to a Line
13.5.2 Unit Tangent and Normal Vectors to a Parabola
13.5.3 Unit Tangent and Normal Vectors to a Circle
13.5.4 Unit Tangent and Normal Vectors to an Ellipse
13.5.5 Unit Tangent and Normal Vectors to a Sine Curve
13.5.6 Unit Tangent and Normal Vectors to a cosh Curve
13.5.7 Unit Tangent and Normal Vectors to a Helix
13.5.8 Unit Tangent and Normal Vectors to a Quadratic Bézier Curve
13.6 Unit Tangent and Normal Vectors to a Surface
13.6.1 Unit Normal Vectors to a Bilinear Patch
13.6.2 Unit Normal Vectors to a Quadratic Bézier Patch
13.6.3 Unit Tangent and Normal Vector to a Sphere
13.6.4 Unit Tangent and Normal Vectors to a Torus
13.7 Summary
13.7.1 Summary of Formulae
14 Continuity
14.1 Introduction
14.2 B-Splines
14.2.1 Uniform B-Splines
14.2.2 B-Spline Continuity
14.3 Derivatives of a Bézier Curve
14.4 Summary
15 Curvature
15.1 Introduction
15.2 Curvature
15.2.1 Curvature of a Circle
15.2.2 Curvature of a Helix
15.2.3 Curvature of a Parabola
15.2.4 Parametric Plane Curve
15.2.5 Curvature of a Graph
15.2.6 Curvature of a 2D Quadratic Bézier Curve
15.2.7 Curvature of a 2D Cubic Bézier Curve
15.3 Summary
15.3.1 Summary of Formulae
16 Conclusion
A Limit of (sinθ)/θ
B Integrating cosnθ
Index