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Contents
Chapter 1.The Tractrix and Similar Curves. 1
1.1 Introduction. 1
1.2 The Classical Tractrix. 1
1.3 The Child and the Toy 3
1.4 The Jogger and the Dog. 6
1.5 Showing the Motions with MATLAB. 12
1.6 Jogger with Constant Velocity. 15
1.7 Using a Moving Coordinate System. 16
1.7.1 Transformation for Jogger/Dog. 18
1.7.2 Transformation for Child/Toy. 20
1.8 Examples. 22
References. 25
Chapter 2.Trajectory of a Spinning Tennis Ball. 27
2.1 Introduction. 27
2.2 MAPLE Solution. 29
2.3 MATLAB Solution. 32
2.4 Simpler Solution for MATLAB 5.35
References. 37
Chapter 3.The Illumination Problem. 39
3.1 Introduction. 39
3.2 Finding the Minimal Illumination Point on a Road 40
3.3 Varying h 2 to Maximize the Illumination. 42
3.4 Optimal Illumination. 45
3.5 Conclusion. 49
References. 49
Chapter 4.Orbits in the Planar Three-Body Problem 51
4.1 Introduction. 51
4.2 Equations of Motion in Physical Coordinates. 52
4.3 Global Regularization. 56
4.4 The Pythagorean Three-Body Problem. 62
4.5 Conclusions. 70
References. 72
Chapter 5.The Internal Field in Semiconductors. 73
5.1 Introduction. 73
5.2 Solving a Nonlinear Poisson Equation Using MAPLE 74
5.3 MATLAB Solution. 75
References. 79
Chapter 6.Some Least Squares Problems. 81
6.1 Introduction. 81
6.2 Fitting Lines, Rectangles and Squares in the Plane 81
6.3 Fitting Hyperplanes. 93
References. 99
Chapter 7.The Generalized Billiard Problem. 101
7.1 Introduction. 101
7.2 The Generalized Reflection Method. 101
7.2.1 Line and Curve Reflection. 102
7.2.2 Mathematical Description. 103
7.2.3 MAPLE Solution. 104
7.3 The Shortest Trajectory Method. 105
7.3.1 MAPLE Solution. 106
7.4 Examples. 106
7.4.1 The Circular Billiard Table. 106
7.4.2 The Elliptical Billiard Table. 110
7.4.3 The Snail Billiard Table. 114
7.4.4 The Star Billiard Table. 114
7.5 Conclusions. 117
References. 119
Chapter 8.Mirror Curves. 121
8.1 The Interesting Waste. 121
8.2 The Mirror Curves Created by MAPLE. 121
8.3 The Inverse Problem. 123
8.3.1 Outflanking Manoeuvre. 123
8.3.2 Geometrical Construction of a Point on the Pattern Curve. 124
8.3.3 MAPLE Solution. 125
8.3.4 Analytic Solution. 126
8.4 Examples. 126
8.4.1 The Circle as the Mirror Curve. 126
8.4.2 The Line as the Mirror Curve. 128
8.5 Conclusions. 129
References. 132
Chapter 9.Smoothing Filters. 133
9.1 Introduction. 133
9.2 Savitzky-Golay Filter. 133
9.2.1 Filter Coefficients. 134
9.2.2 Results. 137
9.3 Least Squares Filter. 138
9.3.1 Lagrange Equations. 139
9.3.2 Zero Finder. 141
9.3.3 Evaluation of the Secular Function. 142
9.3.4 MEX-Files. 144
9.3.5 Results. 148
References. 150
Chapter 10.The Radar Problem. 153
10.1 Introduction. 153
10.2 Converting Degrees into Radians. 154
10.3 Transformation into Geocentric Coordinates. 155
10.4 The Transformations. 158
10.5 Final Algorithm. 160
10.6 Practical Example. 160
References. 162
Chapter 11.Conformal Mapping of a Circle. 163
11.1 Introduction. 163
11.2 Problem Outline. 163
11.3 MAPLE Solution. 164
11.4 MATLAB Solution. 168
References. 170
Chapter 12.The Spinning Top. 171
12.1 Introduction. 171
12.2 Formulation and Basic Analysis of the Solution 173
12.3 The Numerical Solution. 178
References. 180
Chapter 13.The Calibration Problem. 181
13.1 Introduction. 181
13.2 The Physical Model Description. 181
13.3 Approximation by Splitting the Solution. 184
13.4 Conclusions. 189
References. 190
Chapter 14.Heat Flow Problems. 191
14.1 Introduction. 191
14.2 Heat Flow through a Spherical Wall. 191
14.2.1 A Steady State Heat Flow Model. 192
14.2.2 Fourier Model for Steady State. 193
14.2.3 MAPLE Plots. 194
14.3 Non Stationary Heat Flow through an Agriculture Field. 195
14.3.1 MAPLE Plots. 199
References. 199
Chapter 15.Modeling Penetration Phenomena 201
15.1 Introduction. 201
15.2 Short description of the penetration theory. 201
15.3 The Tate-Alekseevskii model. 203
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Index
A
acceleration, 404, 405
air traffic control, 153
algebraic manipulation, 40
algebraic number, 42
algorithmic differentiation, see differentiation
analemma, 392
angle bisector, 410
animation, 406
aphelion, 384
Archimedes, spiral of, 114
asymptota, 400
augmented method, 435
azimuth, 384
B
Babylonic hours, 394-396
ballistic experiment, 212
beacon, DME and VOR, 373, 374
bisection, method of, 333
BLAS routines, 144
Boltzmann constant, 219
Boltzmann statistics, 73
Bose particles, 219
Bose-Einstein integral, 220
boundary condition, 425
boundary value problem, 73, 191, 196
C
calibration, 181
canonical form, 410
canonically conjugated momenta, 56
catenary curve, 423
celestial mechanics, 51
center of gravity, 56, 96, 231, 237
Cholesky decomposition, 252, 256
circle, 4-6, 22, 24, 126, 127, 129, 164, 166, 360, 387, 390
- great, 385, 387, 388, 394
- transformed, 165
circuit, electronic, 311
code optimization, 59, 239
collision, 51, 55
common subexpressions, 59
compression ratio, 230, 233
condition number, 136, 317
conductivity, 191, 194, 199, 302, 303
confidence interval, 380
conic section, 127, 387
continuity
- of global piecewise function, 434
continuous deformation, 122
contour lines, 366, 370
- orthogonality of, 364, 366
- singular points, 362
convergence
- speed, 429
coordinate metrology, 81, 339
coordinate system
- absolute, 153
- astronomical, 384
- Cartesian, 27, 154
- celestial, 385, 388
- cylindric, 233
- ecliptic, 386, 388
- equatorial, 385
- geocentric, 155, 337
- geographic, 154, 155
- horizontal, 384
- inertial, 171
- moving, 17
- polar, 18, 114, 193, 229-232, 238
- rectangular, 114
- relative, 56
- rotated, 126, 336
- transformation, 158
cosine rule, 238
covariance matrix, 452
critical lines, 360, 362
curve
- area enclosed by, 230
- isopotential, 327, 329
- level, 360, 368, 369
- mirror, 121, 122, 129, 132
- parabolic, 122
- perpendicular, 329
- spiral, 335
D
Debye length, 74
declination, 386-388
die forming, 227
differential equation
- complex, 166
- first order, 2, 30, 169, 178
- for contour lines, 359, 362, 364
- linear, 169, 299, 311, 313, 317
- of catenary function, 424
- of motion, 51, 182
- of precession, 178
- second order, 29, 164, 169, 196
- solution
- - analytic, 22, 204, 304, 314
- - elliptic functions, 178
- - numeric, 2, 28, 31, 51, 123, 167, 168, 178
- - ode, 5, 8, 33, 53, 63, 169, 371
- - periodic, 316
- - Runge-Kutta, see Runge-Kutta
- - series, 184
- - stability, 359
- system, 15, 28, 51, 123, 164
- transformation, 17, 18
- zero crossing, 35
differentiation, 58, 165, 175, 283
- automatic, 52, 58, 59
diode, 299-303
Dirichlet boundary conditions, 74
discontinuity, 127, 225, 312, 314
discretization, 412, 418
discretization error, 335
distance, 39, 81, 82, 93, 95, 103
disturbed function, 110
drag force, 27
E
Earth, axis, 388, 391, 396
Earth, satellite, 333, 334
ecliptic, 384, 386, 391
eigenvalue, 299, 315
- analysis, 379
- convergence, 316
- decomposition, 81, 251, 252, 264, 265, 312, 314, 381
- definiteness, 381
- of Hessian, 44
- stability, 318
- trace, 341
- zeros of polynomial, 264, 267
electrical transient, 299
electron, 73
electronic circuit, 299
ellipse, 12, 110, 127, 156, 233
ellipsoid, 154, 156
elliptic functions, 178
envelope, 129
equations of motion, 51, 57, 331
equator, 385
error function, 197
Euclidean space, 191
Euler angles, 172, 173
Euler method, 281, 334
events capability, 8, 9, 35, 367-369
evolution of motion, 55
evolution of social system, 351
exponential sums, 359
- level curves, 360, 369
- zeros, 360, 370
F
family of hyperbolas, 411
Fermi potential, 73
filter, Savitzky-Golay, 133
finite differences, 76, 138
fixed-point iteration, 423, 427
fluid dynamics, 163
fluid, ideal and real, 28, 219
force, drag and Magnus, 27
Fourier analysis, 322
Fourier decomposition, 311
Fourier equation, 191
Fourier law, 192
free knots, 433
friction, 228-230, 233
Frobenius norm, 94, 341
Fubini's law, 235
function piecewise, 400
G
Г function, 368, 371
Gauss-Newton method, 435
Gaussian elimination, 435
geoid, 156
GIF-file, 408
Givens rotation, 143, 144
gnomon point, 386, 387
gnomonic projection, 383, 388
Gröbner bases, 285, 287
gradient, 44, 61
Gram matrix, 253, 257
gravitation, 27, 29, 171, 323, 324, 328, 330, 331, 333, 334, 384
growth of pigs, 443
H
Hamiltonian, 56
Hamiltonian formalism, 51, 56, 58
heat equation, 191, 196
heat flow, 191, 193
Hermite interpolation, 275
Heron's rule, 232
Hessian matrix, 44, 327, 328, 379, 381
Heun, method of, 286
hidden line removal, 97
horizon, 384
hour angle, 385, 387, 388
hours, Babylonic and Italic, 394-396
hyperbola, 127
hyperbola branch, 411
hyperbolic function, 426
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