Geometry for Naval Architects

Cover......Page 1
GEOMETRY
FOR NAVAL
ARCHITECTS
......Page 4
Copyright
......Page 5
Dedication......Page 6
About the Author......Page 7
Preface......Page 8
 The Organization of the Book......Page 10
 Software......Page 11
 Notation......Page 12
Acknowledgements......Page 13
Part 1: Traditional Methods
......Page 14
1 Elements of Descriptive Geometry......Page 15
1.1 Introduction......Page 16
1.3 How We See - The Central Projection......Page 18
1.4.1 Definition......Page 20
1.4.2 Properties......Page 21
1.4.3 Vanishing Points......Page 26
1.5 A Note on Stereoscopic Vision......Page 29
1.6.2 A Few Properties......Page 31
1.6.3 The Concept of Scale......Page 32
1.7.1 Definition......Page 33
1.7.2 The Projection of a Right Angle......Page 35
1.8 The Method of Monge......Page 37
1.9 Points......Page 39
1.10.1 The Projections of a Straight Line......Page 41
1.10.2 Intersecting Lines......Page 43
1.11 Planes......Page 44
1.12 An Example of Plane-Faceted Solid - The Cube......Page 47
1.13 A Space Curve - The Helix......Page 49
1.14 The Cylinder......Page 50
1.15.1 Introduction......Page 53
1.15.2 Points on the Cone Surface......Page 54
1.16.1 Introduction......Page 56
1.16.2 The Circle......Page 57
1.16.3 The Ellipse......Page 58
1.16.4 The Parabola......Page 60
1.16.5 The Hyperbola......Page 62
1.17 What Is Axonometry......Page 64
1.17.1 The Law of Scales......Page 66
1.17.2 Isometry......Page 68
1.17.3 An Ambiguity of the Isometric Projection......Page 72
1.18.1 What Is a Developed Surface......Page 73
1.18.2 The Development of a Cylindrical Surface......Page 74
1.18.3 The Development of a Conic Surface......Page 75
1.19 Summary......Page 76
1.20 Exercises......Page 78
Appendix 1.A The Connection to Linear Algebra and MATLAB......Page 82
Appendix 1.B First Steps in MultiSurf......Page 87
2.1 Introduction......Page 93
2.2.1 A Simple, Idealized Hull Surface......Page 98
2.3 Main Dimensions and Coefficients of Form......Page 101
2.4 Systems of Coordinates......Page 106
2.5 The Hull Surface of a Real Ship......Page 107
2.6 Consistency and Fairness of Ship Lines......Page 108
2.7 Drawing Instruments......Page 114
2.8 Table of Offsets......Page 115
2.9 Shell Expansion and Wetted Surface......Page 116
2.10 An Example in MultiSurf......Page 120
2.11 Summary......Page 128
2.12 Exercises......Page 130
3 Geometric Properties of Areas and Volumes......Page 133
3.1 Introduction......Page 134
3.2.1 Translation of Coordinate Axes......Page 135
3.2.2 Rotation of Coordinate Axes......Page 136
3.3.1 Definitions......Page 137
3.3.2 Examples......Page 139
3.3.3 Examples in Naval Architecture......Page 142
3.4.1 Definitions......Page 143
3.4.2 Examples......Page 144
3.4.3 Examples in Naval Architecture......Page 145
3.5.1 Definitions......Page 146
3.5.2 Parallel Translation of Axes......Page 148
3.5.3 Rotation of Axes......Page 149
3.5.4 The Tensor of Inertia......Page 152
3.5.5 Radius of Gyration......Page 153
3.5.6 The Ellipse of Inertia......Page 154
3.5.7 A Problem of Eigenvalues......Page 155
3.5.8 Examples......Page 158
3.5.9 Examples in Naval Architecture......Page 167
3.6.2 Examples......Page 168
3.6.3 Moments and Centroids of Volumes......Page 171
3.7 Mass Properties......Page 173
3.8 Green\'s Theorem......Page 175
3.9 Hull Transformations......Page 181
3.9.1 Numerical Calculations......Page 182
3.9.2 The `One Minus Prismatic\' Method......Page 184
3.9.3 Swinging the Curve......Page 186
3.9.4 Lackenby\'s General Method......Page 188
3.10.1 The planimeter......Page 189
3.10.2 A MATLAB Digitizer......Page 194
3.11 Summary......Page 195
3.12 Exercises......Page 200
Part 2: Differential Geometry
......Page 207
4.1 Introduction......Page 208
4.2 Parametric Representation......Page 209
4.3 Parametric Equation of Straight Line......Page 212
4.4.1 The Straight Line......Page 215
4.4.2 Working With Parametric Equations......Page 216
4.5 Derivatives of Parametric Functions......Page 218
4.6 Notation of Derivatives......Page 220
4.8 Arc Length......Page 221
4.9 Arc-Length Parametrization......Page 222
4.10.1 Parametric Equations......Page 224
4.10.2 A Theorem on the Axis of Inclination......Page 227
4.10.4 Parametric Equations for Small Angles of Inclination......Page 228
4.11 Summary......Page 230
4.12 Exercises......Page 231
5.1 Introduction......Page 234
5.2 The Definition of Curvature......Page 235
5.2.1 Curvature in Explicit Representation......Page 236
5.2.2 Curvature in Parametric Representation......Page 237
5.3.2 Definition 1 detailed......Page 238
5.3.3 Definition 2 Detailed......Page 239
5.3.4 Definition 3 Detailed......Page 241
5.3.5 Centre of Curvature in Parametric Representation......Page 242
5.4.1 Position......Page 244
5.4.2 Velocity......Page 245
5.4.3 Acceleration......Page 246
5.5 Another Application in Mechanics - The Elastic Line......Page 247
5.6 An Application in Naval Architecture - The Metacentric Radius......Page 249
5.8 Curves in Space......Page 250
5.9 Evolutes......Page 252
5.10 A Lemma on the Normal to a Curve in Implicit Form......Page 253
5.11 Envelopes......Page 255
5.12 The Metacentric Evolute......Page 257
5.14 Examples......Page 260
5.15 Summary......Page 263
5.16 Exercises......Page 265
Appendix 5.A Curvature in MultiSurf......Page 266
6.1 Introduction......Page 269
6.2 Parametric Representation......Page 270
6.3 Curves on Surfaces......Page 276
6.4 First Fundamental Form......Page 277
6.5 Second Fundamental Form......Page 280
6.6 Principal, Gaussian, and Mean Curvatures......Page 285
6.7 Ruled Surfaces......Page 288
6.7.1 Cylindrical Surfaces......Page 289
6.7.2 Conic Surfaces......Page 290
6.7.3 Surfaces of Tangents......Page 291
6.7.4 A Doubly-Ruled Surface, the Hyperboloid of One Sheet......Page 292
6.8 Geodesic Curvature......Page 293
6.9 Developable Surfaces......Page 295
6.10 Geodesics and Plate Development......Page 298
6.11 On the Nature of Surface Curvature......Page 300
6.12 Summary......Page 303
6.13 Exercises......Page 306
Appendix 6.A A Few MultiSurf Tools for Working With Surfaces......Page 308
Part 3: Computer Methods
......Page 313
7.1 Introduction......Page 314
7.2 Cubic Splines......Page 316
7.3 The MATLAB Spline......Page 317
7.4 Working With Parametric Splines......Page 319
7.5 Space Curves......Page 321
7.6 Chord-Length Parametrization......Page 323
7.7 Centripetal Parametrization......Page 325
7.8 Summary......Page 326
7.9 Exercises......Page 327
Appendix 7.A MultiSurf - Cubic Spline, Polycurve......Page 330
8.1 Introduction......Page 334
8.2.1 Translation......Page 337
8.2.2 Rotation Around the Origin......Page 338
8.2.3 Rotation About an Arbitrary Point......Page 339
8.2.5 Isometries......Page 341
8.2.7 Scaling About the Origin......Page 343
8.2.8 Affine Transformations......Page 344
8.2.9 Homogeneous Coordinates......Page 346
8.3 Transformations in 3D Space......Page 349
8.4.1 The Projection Matrix......Page 351
8.4.2 Ideal and Vanishing Points......Page 354
8.4.4 The Orthographic Projection as Limit of Perspective Projection......Page 356
8.5.1 Affine Combination of Two Points - Collinearity......Page 357
8.5.2 Alternative Proof of Collinearity......Page 359
8.5.3 Affine Combination of Three Points - Coplanarity......Page 360
8.6 Barycentres......Page 362
8.7 Summary......Page 363
8.8 Exercises......Page 365
9.1 Introduction......Page 370
9.3 The Second-Degree Bézier Curves......Page 372
9.4 The Third-Degree Bézier Curves......Page 373
9.5 The General Definition of Bézier Curves......Page 374
9.6 Interactive Manipulation of Bézier Curves......Page 376
9.7 De Casteljau\'s Algorithm......Page 377
9.8.1 The First and the Last Point of the Curve......Page 380
9.8.3 Convex Hull......Page 381
9.8.5 Invariance Under Affine Transformations......Page 382
9.9 Joining Two Bézier Curves......Page 384
9.11 Rational Bézier Curves......Page 385
9.12 Summary......Page 389
9.13 Exercises......Page 390
10.1 Introduction......Page 395
10.2 B-Splines......Page 396
10.3 Quadratic B-Splines......Page 397
10.5 A Cubic B-Spline......Page 400
10.6 Phantom Points......Page 402
10.7 Some Properties of the B-Splines......Page 403
10.8 NURBS......Page 405
10.9 Summary......Page 411
10.10 Exercises......Page 413
Appendix 10.A A Note on B-Splines and NURBS in MultiSurf......Page 416
11.1 Introduction......Page 418
11.2 Bézier Patches......Page 419
11.2.1 A Bilinear Patch......Page 420
11.2.2 Curve on Surface......Page 426
11.3 Bicubic Bézier Patch......Page 427
11.4 Joining Two Bézier Patches......Page 432
11.5 Swept Surfaces......Page 435
11.6 Lofted Surfaces......Page 437
11.7 Computer-Aided Design of Hull Surfaces......Page 440
11.8 Summary......Page 442
11.9 Exercises......Page 443
Appendix 11.A A Note on Surfaces in MultiSurf......Page 445
Part 4: Applications in Naval Architecture
......Page 446
12.1 Introduction......Page 447
12.2 Affine Hulls......Page 448
12.4 Affine Combinations of Offsets......Page 452
12.5 Morphing......Page 453
12.7 Summary......Page 456
12.8 Exercises......Page 457
13.1 Introduction......Page 459
13.2 Working With Complex Variables......Page 460
13.3 Conformal Mapping......Page 463
13.4 Lewis Forms......Page 465
13.5 Summary......Page 474
13.6 Exercises......Page 475
Bibliography......Page 476
Answers to Selected Exercises......Page 484
Index......Page 499
Back Cover......Page 504