GRAPHICAL METHODS IN APPLIED MATHEMATICS
1 Transformations of the Plane 1 11 Introduction 1 12 Translations 5 13 Scaling about the Origin 6 14 Reflections 8 15 Rotation about the Origin 9 16 Shears 11 17 Concatenation of Transformations 13 18 Applications 15 181 Instancing 15 182 Robotics 17 2 Homogeneous Coordinates and Transformations of the Plane 19 21 Introduction 19 211 Homogeneous Coordinates 21 22 Points at Infinity 23 23 Visualization of the Projective Plane 24 231 Line Model of the Projective Plane 24 232 Spherical Model of the Projective Plane 26 24 Transformations in Homogeneous Coordinates 27 241 Translations 27 242 Scaling about the Origin 28 243 Rotation about the Origin 29 25 Concatenation of Transformations 30 251 Inverse Transformations 31 252 Rotation about an Arbitrary Point 33 253 Reflection in an Arbitrary Line 34 26 Applications 36 261 Instancing 36 262 Device Coordinate Transformation 37 27 Point and Line Geometry in Homogeneous Coordinates 38 3 Homogeneous Coordinates and Transformations of Space 41 31 Homogeneous Coordinates 41 32 Transformations of Space 42 321 Translations 42 322 Scalings and Reflections 43 323 Rotations about the Coordinate Axes 43 324 Rotation about an Arbitrary Line 45 325 Reflection in an Arbitrary Plane 47 33 Applications 49 331 Computer-aided Design 49 332 Orientation of a Rigid Body 50 34 Geometric Methods for Lines and Planes in Space 52 35 Quaternions 56 4 Projections and the Viewing Pipeline 67 41 Introduction 67 42 Projections of the Plane 68 43 Projections of Three-dimensional Space 72 44 The Viewplane Coordinate Mapping 76 45 The Viewing Pipeline 80 46 Classification of Projections 85 461 Classification of Parallel Projections 85 462 Classification of Perspective Projections 90 5 Curves 95 51 Introduction 95 52 Curve Rendering 98 53 Parametric Curves 99 54 Arclength and Reparametrization 102 55 Application: Numerical Controlled Machining and Offsets 107 56 Conics 109 561 Classification of Conics 112 562 Conics in Standard Form 116 563 Intersections of a Conic with a Line 121 564 Parametrization of an Irreducible Conic 124 565 Converting from Parametric Form to Implicit Form 127 57 Conics in Space 130 58 Applications of Conics 132 6 B´ezier Curves I 135 61 Introduction 135 62 B´ezier Curves of Low Degree 136 621 Linear B´ezier Curves 136 622 Quadratic B´ezier Curves 136 623 Cubic B´ezier Curves 137 63 The Effect of Adjusting a Control Point 140 64 The General B´ezier Curve 141 65 Properties of the Bernstein Polynomials 144 66 Convex Hulls 146 67 Properties of B´ezier Curves 147 68 The de Casteljau Algorithm 151 69 Subdivision of a B´ezier Curve 154 610 Applications 157 6101 Rendering 157 6102 Intersection of a Planar B´ezier Curve and a Line 158 6103 Intersection of Two B´ezier Curves 159 7 B´ezier Curves II 161 71 Spatial B´ezier Curves 161 72 Derivatives of B´ezier Curves 162 73 Conversions Between Representations 166 74 Piecewise B´ezier Curves 168 75 Rational B´ezier Curves 175 751 Properties of Rational B´ezier Curves 177 752 de Casteljau Algorithm for Rational Curves 180 753 Projections of Rational B´ezier Curves 181 754 Derivatives of Rational B´ezier Curves 185 8 B-splines 187 81 Integral B-spline Curves 187 811 Properties of the B-spline Curve 194 812 B-spline Types 196 813 Application: Font Design 203 814 Application: Morphing or Soft Object Animation 203 815 The de Boor Algorithm 205 816 Derivatives of a B-spline 207 82 Non-uniform Rational B-Splines (NURBS) 212 821 Projections of NURBS Curves 214 822 Derivatives of NURBS 216 823 Rational de Boor Algorithm 218 83 Knot Insertion 221 9 Surfaces 225 91 Introduction 225 92 Quadric Surfaces 228 921 Offset Surfaces 232 93 B´ezier and B-spline Surfaces 234 931 Properties of B´ezier and B-spline Surfaces 235 932 Derivatives of B´ezier and B-spline Surfaces 238 94 Surface Constructions 241 941 Extruded Surfaces 241 942 Ruled Surfaces 242 943 Translationally Swept Surfaces 244 944 Surfaces of Revolution 245 95 Surface Subdivision 248 96 Skin and Loft Surfaces 251 97 Geometric Modelling and CAD 260 971 Wireframe Modeller 261 972 Surface Modeller 261 973 Constructive Solid Geometry (CSG) Modellers 261 974 Boundary Representations (B-rep) 263 10 Curve and Surface Curvatures 267 101 Curvature of a Plane Curve 267 102 Curvature and Torsion of a Space Curve 275 103 Curvature of B´ezier Curves 283 104 Surface Curvatures 285 11 Rendering 297 111 Introduction 297 112 Colour 298 113 An Illumination Model for Reflected Light 299 1131 Diffuse Reflection 300 1132 Specular Reflection 302 1133 Ambient Reflection 304 1134 Attenuation 305 1135 Total Intensity 305 114 Shading Algorithms 306 1141 Flat Shading 307 1142 Gouraud Shading 307 1143 Phong Shading 309 115 Silhouettes 309 116 Shadows 320 Solutions 323 References 345