Modeling of Curves and Surfaces with MATLAB® (Springer Undergraduate Texts in Mathematics and Technology Book 7)

This text on geometry is devoted to various central geometrical topics including: graphs of functions, transformations, (non-)Euclidean geometries, curves and surfaces as well as their applications in a variety of disciplines. This book presents elementary methods for analytical modeling and demonstrates the potential for symbolic computational tools to support the development of analytical solutions. The author systematically examines several powerful tools of MATLAB® including 2D and 3D animation of geometric images with shadows and colors, transformations using matrices, and then studies more complex geometrical modeling problems related to analysis of curves and surfaces. With over 150 stimulating exercises and problems, this text integrates traditional differential and non-Euclidean geometries with more current computer systems in a practical and user-friendly format. This text greatly extends the author’s previous title, Geometry of Curves and Surfaces with Maple (Birkhäuser, © 2000), and has a different focus. In addition to being applications driven and motivated by numerous examples and exercises from real-world fields, the book also contains over 60 percent new material, including new sections with complex numbers, quaternions, matrices and transformations, hyperbolic geometry, fractals, and surface-splines and over 300 figures reproducible using MATLAB® programs. This text is an excellent classroom resource or self-study reference for undergraduate students in a variety of disciplines, engineers, computer scientists, and instructors of applied mathematics. Cover 1 Springer Undergraduate Texts in Mathematics and Technology 2 Modeling of Curves and Surfaces with MATLAB 4 Copyright 5 0387712771 5 Dedication 6 Foreword 8 Preface 10 Contents 14 Part I Functions and Transformations 18 1 Functions and Graphs 20 1.1 Numbers 20 1.1.1 Integers, rationals, and reals 21 1.1.2 Complex numbers 26 1.1.3 Quaternions 29 1.2 Elementary and Special Functions 31 1.2.1 Plotting in two dimensions 32 1.2.2 Plotting graphs 35 1.2.3 Elementary functions 39 1.2.4 Special functions 43 1.3 Piecewise Functions 45 1.3.1 Piecewise constant and linear functions 46 1.3.2 The functions max and min 48 1.3.3 Functions containing the operation abs 49 1.3.4 Functions that are defined using limit 50 1.3.5 Statistical functions 51 1.4 Graphs in Polar Coordinates 52 1.4.1 Basic plots in polar coordinates 52 1.4.2 Remarkable curves 56 1.4.3 Spirals 58 1.4.4 Roses and crosses 60 1.5 Polynomial Interpolation of Functions 61 1.5.1 Representation of polynomials 62 1.5.2 Lagrange polynomials 66 1.5.3 Hermite polynomial 68 1.5.4 Least-squares analysis 70 1.6 Spline Interpolation of Functions 72 1.6.1 Cubic spline 72 1.6.2 Construction of curves using splines 74 1.6.3 Extremal property of a cubic spline 75 1.7 Two-Dimensional Interpolation and Smoothing 78 1.7.1 Two-dimensional interpolation 79 1.7.2 Two-dimensional smoothing 81 1.7.3 Extremal property of a bicubic spline 82 1.8 Exercises 85 2 Rigid Motions (Isometries) 100 2.1 Vectors 100 2.1.1 Vectors in bold0mu mumu RRdottedRRRR3 100 2.1.2 Lines and conics 102 2.2 Rigid Motions in Rn 104 2.2.1 Rigid motions in two dimensions 104 2.2.2 Rotations and reflections in planes of R3 107 2.2.3 Rotations and reflections using quaternions 111 2.3 Sphere 114 2.3.1 Spherical transformations 115 2.3.2 Stereographic projection 116 2.4 Polyhedra 118 2.4.1 What is a polyhedron? 119 2.4.2 Platonic solids 123 2.4.3 Symmetries of regular polygons and polyhedra 126 2.4.4 Star-shaped polyhedra 129 2.4.5 Archimedean solids 133 2.5 Appendix: Matrices and Groups 138 2.5.1 Permutations and group actions 138 2.5.2 Matrices and linear transformations 140 2.6 Exercises 143 3 Affine and Projective Transformations 152 3.1 Affine Transformations 152 3.1.1 Affine transformations in two and three dimensions 152 3.1.2 Parallel projections 156 3.1.3 Convex hull and Delaunay triangulation 158 3.2 Projective Transformations 161 3.2.1 Homogeneous coordinates 161 3.2.2 Duality and cross-ratio 162 3.2.3 Projective transformations of RP2 164 3.3 Transformations in Homogeneous Coordinates 167 3.3.1 Transformations in two dimensions 168 3.3.2 Transformations in three dimensions 170 3.4 Exercises 171 4 Möbius Transformations 176 4.1 Inversive Geometry 176 4.1.1 Reflection in a sphere 176 4.1.2 Inversive geometry of a plane 178 4.2 Möbius Transformations 180 4.2.1 Möbius transformations in two dimensions 180 4.2.2 Möbius transformations in three dimensions 185 4.2.3 The sphere and Möbius transformations 186 4.3 Hyperbolic Geometry 187 4.3.1 The hyperbolic length 187 4.3.2 The hyperbolic distance 190 4.3.3 The hyperbolic area 192 4.3.4 Hyperbolic motions in two dimensions 194 4.3.5 Hyperbolic motions in three dimensions 196 4.4 Examples of Visualization 199 4.5 Exercises 208 Part II Curves and Surfaces 216 5 Examples of Curves 218 5.1 Plane Curves 218 5.1.1 Parametric plane curves 219 5.1.2 Level lines and extremal problems 223 5.1.3 Trajectories of a vector field and ODEs 227 5.1.4 Euler's equations 230 5.2 Fractal Curves 233 5.2.1 Peano curves 233 5.2.2 Koch snowflake 235 5.2.3 Sierpinski's carpet 237 5.2.4 Menger cube 238 5.3 Space Curves and Projections 239 5.3.1 Intuitive projection 239 5.3.2 Oblique and axonometric projections 243 5.4 More Examples of Space Curves 245 5.4.1 Springs on surfaces 245 5.4.2 Curves as intersections of surfaces 247 5.4.3 Curves with shadows 249 5.4.4 Construction of curves using polynomials 252 5.5 Exercises 255 6 Geometry of Curves 262 6.1 Tangent Lines 262 6.1.1 Tangent lines, normals, and the Frenet frame 262 6.1.2 Asymptotes of curves 267 6.1.3 Envelope of a family of curves 270 6.1.4 Mathematical embroidery 272 6.1.5 Evolutes and evolvents, and parallel curves 275 6.2 Singular Points 279 6.2.1 Singular points of parametric curves 280 6.2.2 Singular points of implicitly defined curves 281 6.2.3 Unusual singular points of plane curves 283 6.3 Length and Center of Mass 284 6.4 Curvature and Torsion 289 6.4.1 Basic formulae 290 6.4.2 Curvature of a plane curve 291 6.4.3 Curvature and torsion of a space curve 293 6.5 Exercises 296 7 Geometry of Surfaces 302 7.1 Regular Surface 302 7.1.1 What is a surface? 302 7.1.2 Regular surfaces 306 7.1.3 Methods of generating surfaces 307 7.2 Tangent Planes and Normal Vectors 311 7.2.1 Basic formulae and properties 311 7.2.2 Extrema of functions defined on surfaces 314 7.3 Singular Points on Surfaces 316 7.3.1 Parametric surfaces 316 7.3.2 Implicitly given surfaces 318 7.4 Osculating Paraboloid 321 7.4.1 Properties of the osculating paraboloid 321 7.4.2 Plotting an osculating paraboloid 324 7.5 Curvature and Geodesics 327 7.5.1 Gaussian and mean curvatures 327 7.5.2 Geodesics 329 7.6 Exercises 331 8 Examples of Surfaces 340 8.1 Surfaces of Revolution 340 8.2 Algebraic Surfaces 346 8.3 Ruled Surfaces 348 8.4 Envelope of a One-Parameter Family of Surfaces 354 8.5 More Examples of Surfaces 357 8.5.1 Canal surfaces and tubes 357 8.5.2 Translation surfaces 358 8.5.3 Twisted surfaces 360 8.5.4 Parallel surfaces 360 8.5.5 Pedal and podoid surfaces 362 8.5.6 Cissoidal and conchoidal maps 363 8.5.7 Inversion of a surface 365 8.6 Exercises 367 9 Piecewise Curves and Surfaces 374 9.1 Introduction 374 9.2 Bézier Curves 376 9.2.1 Bézier curves of degree n 376 9.2.2 Piecewise cubic Bézier curves 379 9.2.3 Rational Bézier curves 381 9.3 Hermite Interpolation Curves 383 9.3.1 Cubic Hermite curves 383 9.3.2 Piecewise Hermite curves 384 9.3.3 The cubic spline interpolation 387 9.3.4 Application: Cardinal spline curves 388 9.4 -Spline Curves 389 9.4.1 Geometrical continuity of curves 390 9.4.2 Cubic -spline curve 395 9.4.3 -Spline curves in space 398 9.4.4 B-spline curves 400 9.5 The Cartesian Product Surface Patch 404 9.6 Bicubic Spline Surfaces 411 9.6.1 Double vertices 412 9.6.2 Triple vertices 414 9.6.3 Phantom vertices 415 9.6.4 Examples 416 9.7 Exercises 417 Appendix M-Files 430 A.1 Polynomial Interpolation 430 A.2 Bézier Curves 431 A.3 Hermite Curves 432 A.4 -Spline Curves 433 A.5 Spline Surfaces 436 A.6 Semi-Regular Polyhedra 437 A.7 Hyperbolic Geometry 445 A.8 Investigation of a Surface in R3 455 References 458 Index 464 9780387712772 This text on geometry is devoted to various central geometrical topics including: graphs of functions, transformations, (non- )Euclidean geometries, curves and surfaces as well as their applications in a variety of disciplines. This book presents elementary methods for analytical modeling and demonstrates the potential for symbolic computational tools to support the development of analytical solutions. The author systematically examines several powerful tools of MATLAB® including 2D and 3D animation of geometric images with shadows and colors, transformations using matrices, and then studies more complex geometrical modeling problems related to analysis of curves and surfaces. With over 150 stimulating exercises and problems, this text integrates traditional differential and non-Euclidean geometries with more current computer systems in a practical and user-friendly format. This text greatly extends the author's previous title, Geometry of Curves and Surfaces with Maple (Birkhäuser, c2000), and has a different focus. In addition to being applications driven and motivated by numerous examples and exercises from real-world fields, the book also contains over 60 percent new material, including new sections with complex numbers, quaternions, matrices and transformations, hyperbolic geometry, fractals, and surface-splines and over 300 figures reproducible using MATLAB® programs. This text is an excellent classroom resource or self-study reference for undergraduate students in a variety of disciplines, engineers, computer scientists, and instructors of applied mathematics This text on geometry is devoted to various central geometrical topics including: graphs of functions, transformations, (non-)Euclidean geometries, curves and surfaces as well as their applications in a variety of disciplines. This book presents elementary methods for analytical modeling and demonstrates the potential for symbolic computational tools to support the development of analytical solutions. The author systematically examines several powerful tools of MATLAB® including 2D and 3D animation of geometric images with shadows and colors and transformations using matrices. With over 150 stimulating exercises and problems, this text integrates traditional differential and non-Euclidean geometries with more current computer systems in a practical and user-friendly format. This text is an excellent classroom resource or self-study reference for undergraduate students in a variety of disciplines. Front Matter....Pages i-xv Front Matter....Pages 1-1 Functions and Graphs....Pages 3-82 Rigid Motions (Isometries)....Pages 83-134 Affine and Projective Transformations....Pages 135-158 Möbius Transformations....Pages 159-197 Front Matter....Pages 200-200 Examples of Curves....Pages 201-244 Geometry of Curves....Pages 245-284 Geometry of Surfaces....Pages 285-321 Examples of Surfaces....Pages 323-356 Piecewise Curves and Surfaces....Pages 357-411 Back Matter....Pages 413-452