Calculus for Computer Graphics [2nd ed.]

Preface......Page 3
Contents......Page 6
1.1 What is Calculus?......Page 13
1.3 Who Invented Calculus?......Page 14
2.2 Expressions, Variables, Constants and Equations......Page 16
2.3 Functions......Page 17
2.3.1 Continuous and Discontinuous Functions......Page 18
2.3.2 Linear Functions......Page 19
2.3.4 Polynomial Functions......Page 20
2.3.6 Other Functions......Page 21
2.4.1 Slope of a Function......Page 22
2.4.2 Differentiating Periodic Functions......Page 26
2.5 Summary......Page 28
3.1 Introduction......Page 30
3.2 Small Numerical Quantities......Page 31
3.3.1 Quadratic Function......Page 32
3.3.2 Cubic Equation......Page 33
3.3.3 Functions and Limits......Page 35
3.3.4 Graphical Interpretation of the Derivative......Page 37
3.3.5 Derivatives and Differentials......Page 38
3.3.6 Integration and Antiderivatives......Page 39
3.4 Summary......Page 41
3.5.2 Limiting Value of a Quotient......Page 42
3.5.4 Slope of a Polynomial......Page 43
3.5.6 Integrate a Polynomial......Page 44
4.1 Introduction......Page 45
4.2.1 Sums of Functions......Page 46
4.2.2 Function of a Function......Page 48
4.2.3 Function Products......Page 51
4.2.4 Function Quotients......Page 54
4.3 Differentiating Implicit Functions......Page 57
4.4.1 Exponential Functions......Page 60
4.4.2 Logarithmic Functions......Page 62
4.5 Differentiating Trigonometric Functions......Page 64
4.5.1 Differentiating tan......Page 65
4.5.3 Differentiating sec......Page 66
4.5.5 Differentiating arcsin, arccos and arctan......Page 68
4.5.6 Differentiating arccsc, arcsec and arccot......Page 69
4.5.7 Summary: Trigonometric Functions......Page 70
4.6 Differentiating Hyperbolic Functions......Page 72
4.6.1 Differentiating sinh, cosh and tanh......Page 73
4.6.2 Differentiating cosech, sech and coth......Page 75
4.6.3 Differentiating arsinh, arcosh and artanh......Page 76
4.6.4 Differentiating arcsch, arsech and arcoth......Page 77
4.7 Summary......Page 80
5.2 Higher Derivatives of a Polynomial......Page 81
5.3 Identifying a Local Maximum or Minimum......Page 83
5.4 Derivatives and Motion......Page 86
5.5.1 Summary of Formulae......Page 89
6.2 Partial Derivatives......Page 90
6.2.1 Visualising Partial Derivatives......Page 93
6.2.2 Mixed Partial Derivatives......Page 95
6.3 Chain Rule......Page 97
6.4 Total Derivative......Page 99
6.6.1 Summary of Formulae......Page 100
7.2 Indefinite Integral......Page 101
7.3 Standard Integration Formulae......Page 102
7.4.1 Continuous Functions......Page 103
7.4.2 Difficult Functions......Page 104
7.4.3 Trigonometric Identities......Page 105
7.4.4 Exponent Notation......Page 107
7.4.5 Completing the Square......Page 108
7.4.6 The Integrand Contains a Derivative......Page 110
7.4.7 Converting the Integrand into a Series of Fractions......Page 112
7.4.8 Integration by Parts......Page 113
7.4.9 Integration by Substitution......Page 121
7.4.10 Partial Fractions......Page 125
7.5 Summary......Page 128
8.2 Calculating Areas......Page 129
8.3 Positive and Negative Areas......Page 137
8.4 Area Between Two Functions......Page 138
8.5 Areas with the y-Axis......Page 140
8.6 Area with Parametric Functions......Page 141
8.7.1 Domains and Intervals......Page 143
8.7.2 The Riemann Sum......Page 144
8.8 Summary......Page 145
9.2 Lagrange's Mean-Value Theorem......Page 146
9.3 Arc Length......Page 147
9.3.2 Arc Length of a Circle......Page 149
9.3.3 Arc Length of a Parabola......Page 150
9.3.4 Arc Length of y=x32......Page 154
9.3.6 Arc Length of a Hyperbolic Cosine Function......Page 155
9.3.7 Arc Length of Parametric Functions......Page 156
9.3.8 Arc Length of a Circle......Page 157
9.3.9 Arc Length of an Ellipse......Page 158
9.3.10 Arc Length of a Helix......Page 159
9.3.11 Arc Length of a 2D Quadratic Bézier Curve......Page 160
9.3.12 Arc Length of a 3D Quadratic Bézier Curve......Page 162
9.3.13 Arc Length Parameterisation of a 3D Line......Page 164
9.3.14 Arc Length Parameterisation of a Helix......Page 167
9.3.15 Positioning Points on a Straight Line Using  a Square Law......Page 168
9.3.16 Positioning Points on a Helix Curve Using  a Square Law......Page 169
9.3.17 Arc Length Using Polar Coordinates......Page 171
9.4 Summary......Page 173
References......Page 174
10.2 Surface of Revolution......Page 175
10.2.2 Surface Area of a Right Cone......Page 177
10.2.3 Surface Area of a Sphere......Page 179
10.2.4 Surface Area of a Paraboloid......Page 181
10.3 Surface Area Using Parametric Functions......Page 182
10.4 Double Integrals......Page 184
10.5.1 1D Jacobian......Page 186
10.5.2 2D Jacobian......Page 187
10.5.3 3D Jacobian......Page 192
10.6 Double Integrals for Calculating Area......Page 194
10.7.1 Summary of Formulae......Page 200
11.2 Solid of Revolution: Disks......Page 202
11.2.1 Volume of a Cylinder......Page 203
11.2.2 Volume of a Right Cone......Page 204
11.2.3 Volume of a Right Conical Frustum......Page 205
11.2.4 Volume of a Sphere......Page 207
11.2.5 Volume of an Ellipsoid......Page 208
11.2.6 Volume of a Paraboloid......Page 209
11.3 Solid of Revolution: Shells......Page 210
11.3.1 Volume of a Cylinder......Page 211
11.3.3 Volume of a Sphere......Page 212
11.3.4 Volume of a Paraboloid......Page 214
11.4 Volumes with Double Integrals......Page 215
11.4.2 Rectangular Box......Page 216
11.4.3 Rectangular Prism......Page 217
11.4.4 Curved Top......Page 218
11.4.6 Cylinder......Page 219
11.4.7 Truncated Cylinder......Page 220
11.5.1 Rectangular Box......Page 222
11.5.2 Volume of a Cylinder......Page 223
11.5.3 Volume of a Sphere......Page 225
11.5.4 Volume of a Cone......Page 226
11.6 Summary......Page 227
11.6.1 Summary of Formulae......Page 228
12.2 Differentiating Vector Functions......Page 229
12.2.1 Velocity and Speed......Page 230
12.2.3 Rules for Differentiating Vector-Valued Functions......Page 232
12.3 Integrating Vector-Valued Functions......Page 233
12.3.2 Position of a Moving Object......Page 234
12.4.1 Summary of Formulae......Page 235
13.2 Notation......Page 237
13.3 Tangent Vector to a Curve......Page 238
13.4 Normal Vector to a Curve......Page 240
13.5 Gradient of a Scalar Field......Page 241
13.5.1 Unit Tangent and Normal Vectors to a Line......Page 244
13.5.2 Unit Tangent and Normal Vectors to a Parabola......Page 246
13.5.3 Unit Tangent and Normal Vectors to a Circle......Page 248
13.5.4 Unit Tangent and Normal Vectors to an Ellipse......Page 250
13.5.5 Unit Tangent and Normal Vectors to a Sine Curve......Page 252
13.5.6 Unit Tangent and Normal Vectors to a cosh Curve......Page 253
13.5.7 Unit Tangent and Normal Vectors to a Helix......Page 255
13.5.8 Unit Tangent and Normal Vectors to a Quadratic Bézier Curve......Page 257
13.6.1 Unit Normal Vectors to a Bilinear Patch......Page 259
13.6.2 Unit Normal Vectors to a Quadratic Bézier Patch......Page 260
13.6.3 Unit Tangent and Normal Vector to a Sphere......Page 263
13.6.4 Unit Tangent and Normal Vectors to a Torus......Page 265
13.7.1 Summary of Formulae......Page 267
14.2 B-Splines......Page 270
14.2.1 Uniform B-Splines......Page 271
14.2.2 B-Spline Continuity......Page 272
14.3 Derivatives of a Bézier Curve......Page 274
14.4 Summary......Page 278
15.2 Curvature......Page 279
15.2.1 Curvature of a Circle......Page 280
15.2.2 Curvature of a Helix......Page 281
15.2.3 Curvature of a Parabola......Page 283
15.2.4 Parametric Plane Curve......Page 285
15.2.6 Curvature of a 2D Quadratic Bézier Curve......Page 288
15.3 Summary......Page 290
15.3.1 Summary of Formulae......Page 291
16 Conclusion......Page 293
Limit of (sinθ)/θ......Page 294
Integrating cosnθ......Page 297
Index......Page 299