Calculus for Computer Graphics

Preface......Page 6
Contents......Page 9
 1.1 What is Calculus?......Page 16
 1.3 Who Invented Calculus?......Page 17
 2.2 Expressions, Variables, Constants and Equations......Page 19
 2.3 Functions......Page 20
  2.3.1 Continuous and Discontinuous Functions......Page 21
  2.3.2 Linear Functions......Page 22
  2.3.4 Polynomial Functions......Page 23
  2.3.6 Other Functions......Page 24
  2.4.1 Slope of a Function......Page 25
  2.4.2 Differentiating Periodic Functions......Page 29
 2.5 Summary......Page 31
 3.1 Introduction......Page 33
 3.2 Small Numerical Quantities......Page 34
  3.3.1 Quadratic Function......Page 35
  3.3.2 Cubic Equation......Page 36
  3.3.3 Functions and Limits......Page 38
  3.3.4 Graphical Interpretation of the Derivative......Page 40
  3.3.5 Derivatives and Differentials......Page 41
  3.3.6 Integration and Antiderivatives......Page 42
 3.4 Summary......Page 44
  3.5.2 Limiting Value of a Quotient......Page 45
  3.5.4 Slope of a Polynomial......Page 46
  3.5.6 Integrate a Polynomial......Page 47
 4.1 Introduction......Page 48
  4.2.1 Sums of Functions......Page 49
  4.2.2 Function of a Function......Page 51
  4.2.3 Function Products......Page 54
  4.2.4 Function Quotients......Page 57
 4.3 Differentiating Implicit Functions......Page 60
  4.4.1 Exponential Functions......Page 63
  4.4.2 Logarithmic Functions......Page 65
 4.5 Differentiating Trigonometric Functions......Page 67
  4.5.1 Differentiating tan......Page 68
  4.5.3 Differentiating sec......Page 69
  4.5.5 Differentiating arcsin, arccos and arctan......Page 71
  4.5.6 Differentiating arccsc, arcsec and arccot......Page 72
  4.5.7 Summary: Trigonometric Functions......Page 73
 4.6 Differentiating Hyperbolic Functions......Page 75
  4.6.1 Differentiating sinh, cosh and tanh......Page 76
  4.6.2 Differentiating cosech, sech and coth......Page 78
  4.6.3 Differentiating arsinh, arcosh and artanh......Page 79
  4.6.4 Differentiating arcsch, arsech and arcoth......Page 80
 4.7 Summary......Page 83
 5.2 Higher Derivatives of a Polynomial......Page 84
 5.3 Identifying a Local Maximum or Minimum......Page 86
 5.4 Derivatives and Motion......Page 89
  5.5.1 Summary of Formulae......Page 92
 6.2 Partial Derivatives......Page 93
  6.2.1 Visualising Partial Derivatives......Page 96
  6.2.2 Mixed Partial Derivatives......Page 98
 6.3 Chain Rule......Page 100
 6.4 Total Derivative......Page 102
  6.6.1 Summary of Formulae......Page 103
 7.2 Indefinite Integral......Page 104
 7.3 Standard Integration Formulae......Page 105
  7.4.1 Continuous Functions......Page 106
  7.4.2 Difficult Functions......Page 107
  7.4.3 Trigonometric Identities......Page 108
  7.4.4 Exponent Notation......Page 110
  7.4.5 Completing the Square......Page 111
  7.4.6 The Integrand Contains a Derivative......Page 113
  7.4.7 Converting the Integrand into a Series of Fractions......Page 115
  7.4.8 Integration by Parts......Page 116
  7.4.9 Integration by Substitution......Page 124
  7.4.10 Partial Fractions......Page 128
 7.5 Summary......Page 131
 8.2 Calculating Areas......Page 132
 8.3 Positive and Negative Areas......Page 140
 8.4 Area Between Two Functions......Page 141
 8.5 Areas with the y-Axis......Page 143
 8.6 Area with Parametric Functions......Page 144
  8.7.1 Domains and Intervals......Page 146
  8.7.2 The Riemann Sum......Page 147
 8.8 Summary......Page 148
 9.2 Lagrange's Mean-Value Theorem......Page 149
 9.3 Arc Length......Page 150
  9.3.2 Arc Length of a Circle......Page 152
  9.3.3 Arc Length of a Parabola......Page 153
  9.3.4 Arc Length of y=x32......Page 157
  9.3.6 Arc Length of a Hyperbolic Cosine Function......Page 158
  9.3.7 Arc Length of Parametric Functions......Page 159
  9.3.8 Arc Length of a Circle......Page 160
  9.3.9 Arc Length of an Ellipse......Page 161
  9.3.10 Arc Length of a Helix......Page 162
  9.3.11 Arc Length of a 2D Quadratic Bézier Curve......Page 163
  9.3.12 Arc Length of a 3D Quadratic Bézier Curve......Page 165
  9.3.13 Arc Length Parameterisation of a 3D Line......Page 167
  9.3.14 Arc Length Parameterisation of a Helix......Page 170
  9.3.15 Positioning Points on a Straight Line Using  a Square Law......Page 171
  9.3.16 Positioning Points on a Helix Curve Using  a Square Law......Page 172
  9.3.17 Arc Length Using Polar Coordinates......Page 174
 9.4 Summary......Page 176
 References......Page 177
 10.2 Surface of Revolution......Page 178
  10.2.2 Surface Area of a Right Cone......Page 180
  10.2.3 Surface Area of a Sphere......Page 182
  10.2.4 Surface Area of a Paraboloid......Page 184
 10.3 Surface Area Using Parametric Functions......Page 185
 10.4 Double Integrals......Page 187
  10.5.1 1D Jacobian......Page 189
  10.5.2 2D Jacobian......Page 190
  10.5.3 3D Jacobian......Page 195
 10.6 Double Integrals for Calculating Area......Page 197
  10.7.1 Summary of Formulae......Page 203
 11.2 Solid of Revolution: Disks......Page 205
  11.2.1 Volume of a Cylinder......Page 206
  11.2.2 Volume of a Right Cone......Page 207
  11.2.3 Volume of a Right Conical Frustum......Page 208
  11.2.4 Volume of a Sphere......Page 210
  11.2.5 Volume of an Ellipsoid......Page 211
  11.2.6 Volume of a Paraboloid......Page 212
 11.3 Solid of Revolution: Shells......Page 213
  11.3.1 Volume of a Cylinder......Page 214
  11.3.3 Volume of a Sphere......Page 215
  11.3.4 Volume of a Paraboloid......Page 217
 11.4 Volumes with Double Integrals......Page 218
  11.4.2 Rectangular Box......Page 219
  11.4.3 Rectangular Prism......Page 220
  11.4.4 Curved Top......Page 221
  11.4.6 Cylinder......Page 222
  11.4.7 Truncated Cylinder......Page 223
  11.5.1 Rectangular Box......Page 225
  11.5.2 Volume of a Cylinder......Page 226
  11.5.3 Volume of a Sphere......Page 228
  11.5.4 Volume of a Cone......Page 229
 11.6 Summary......Page 230
  11.6.1 Summary of Formulae......Page 231
 12.2 Differentiating Vector Functions......Page 232
  12.2.1 Velocity and Speed......Page 233
  12.2.3 Rules for Differentiating Vector-Valued Functions......Page 235
 12.3 Integrating Vector-Valued Functions......Page 236
  12.3.2 Position of a Moving Object......Page 237
  12.4.1 Summary of Formulae......Page 238
 13.2 Notation......Page 240
 13.3 Tangent Vector to a Curve......Page 241
 13.4 Normal Vector to a Curve......Page 243
 13.5 Gradient of a Scalar Field......Page 244
  13.5.1 Unit Tangent and Normal Vectors to a Line......Page 247
  13.5.2 Unit Tangent and Normal Vectors to a Parabola......Page 249
  13.5.3 Unit Tangent and Normal Vectors to a Circle......Page 251
  13.5.4 Unit Tangent and Normal Vectors to an Ellipse......Page 253
  13.5.5 Unit Tangent and Normal Vectors to a Sine Curve......Page 255
  13.5.6 Unit Tangent and Normal Vectors to a cosh Curve......Page 256
  13.5.7 Unit Tangent and Normal Vectors to a Helix......Page 258
  13.5.8 Unit Tangent and Normal Vectors to a Quadratic Bézier Curve......Page 260
  13.6.1 Unit Normal Vectors to a Bilinear Patch......Page 262
  13.6.2 Unit Normal Vectors to a Quadratic Bézier Patch......Page 263
  13.6.3 Unit Tangent and Normal Vector to a Sphere......Page 266
  13.6.4 Unit Tangent and Normal Vectors to a Torus......Page 268
  13.7.1 Summary of Formulae......Page 270
 14.2 B-Splines......Page 273
  14.2.1 Uniform B-Splines......Page 274
  14.2.2 B-Spline Continuity......Page 275
 14.3 Derivatives of a Bézier Curve......Page 277
 14.4 Summary......Page 281
 15.2 Curvature......Page 282
  15.2.1 Curvature of a Circle......Page 283
  15.2.2 Curvature of a Helix......Page 284
  15.2.3 Curvature of a Parabola......Page 286
  15.2.4 Parametric Plane Curve......Page 288
  15.2.6 Curvature of a 2D Quadratic Bézier Curve......Page 291
 15.3 Summary......Page 293
  15.3.1 Summary of Formulae......Page 294
16 Conclusion......Page 296
A Limit of (sinθ)/θ......Page 297
B Integrating cosnθ......Page 300
Index......Page 302