| Introduction | p. xi |
| Plane curves | p. 1 |
| Introduction | p. 1 |
| Regular plane curves and their evolutes | p. 9 |
| Curvature | p. 14 |
| Parallels | p. 22 |
| Equivalent parametric curves | p. 26 |
| Unit-speed curves | p. 27 |
| Unit-angular-velocity curves | p. 28 |
| Rhamphoid cusps | p. 29 |
| The determination of circular points | p. 33 |
| The four-vertex theorem | p. 35 |
| Exercises | p. 37 |
| Some elementary geometry | p. 42 |
| Introduction | p. 42 |
| Some linear facts | p. 42 |
| Some bilinear facts | p. 44 |
| Some projective facts | p. 46 |
| Projective curves | p. 46 |
| Spaces of polynomials | p. 48 |
| Inversion and stereographic projection | p. 48 |
| Exercises | p. 49 |
| Plane kinematics | p. 51 |
| Introduction | p. 51 |
| Instantaneous rotations and translations | p. 51 |
| The motion of a plane at t = 0 | p. 52 |
| The inflection circle and Ball point | p. 53 |
| The cubic of stationary curvature | p. 54 |
| Burmester points | p. 58 |
| Rolling wheels | p. 59 |
| Polodes | p. 61 |
| Caustics | p. 62 |
| Exercises | p. 63 |
| The derivatives of a map | p. 67 |
| Introduction | p. 67 |
| The first derivative and C[superscript 1] submanifolds | p. 67 |
| Higher derivatives and C[superscript k] submanifolds | p. 80 |
| The Faa de Bruno formula | p. 83 |
| Exercises | p. 85 |
| Curves on the unit sphere | p. 88 |
| Introduction | p. 88 |
| Geodesic curvature | p. 89 |
| Spherical kinematics | p. 91 |
| Exercises | p. 94 |
| Space curves | p. 95 |
| Introduction | p. 95 |
| Space curves | p. 95 |
| The focal surface and space evolute | p. 100 |
| The Serret--Frenet equations | p. 105 |
| Parallels | p. 107 |
| Close up views | p. 113 |
| Historical note | p. 116 |
| Exercises | p. 116 |
| k-times linear forms | p. 119 |
| Introduction | p. 119 |
| k-times linear forms | p. 119 |
| Quadratic forms on R[superscript 2] | p. 122 |
| Cubic forms on R[superscript 2] | p. 124 |
| Use of complex numbers | p. 129 |
| Exercises | p. 134 |
| Probes | p. 138 |
| Introduction | p. 138 |
| Probes of smooth map-germs | p. 138 |
| Probing a map-germ V: R[superscript 2]--R | p. 141 |
| Optional reading | p. 145 |
| Exercises | p. 151 |
| Contact | p. 152 |
| Introduction | p. 152 |
| Contact equivalence | p. 152 |
| K-equivalence | p. 154 |
| Applications | p. 155 |
| Exercises | p. 156 |
| Surfaces in R[superscript 3] | p. 158 |
| Introduction | p. 158 |
| Euler's formula | p. 167 |
| The sophisticated approach | p. 169 |
| Lines of curvature | p. 172 |
| Focal curves of curvature | p. 173 |
| Historical note | p. 177 |
| Exercises | p. 178 |
| Ridges and ribs | p. 182 |
| Introduction | p. 182 |
| The normal bundle of a surface | p. 182 |
| Isolated umbilics | p. 183 |
| The normal focal surface | p. 184 |
| Ridges and ribs | p. 187 |
| A classification of focal points | p. 189 |
| More on ridges and ribs | p. 191 |
| Exercises | p. 195 |
| Umbilics | p. 198 |
| Introduction | p. 198 |
| Curves through umbilics | p. 199 |
| Classifications of umbilics | p. 201 |
| The main classification | p. 202 |
| Darboux's classification | p. 203 |
| Index | p. 208 |
| Straining a surface | p. 208 |
| The birth of umbilics | p. 210 |
| Exercises | p. 212 |
| The parabolic line | p. 214 |
| Introduction | p. 214 |
| Gaussian curvature | p. 214 |
| The parabolic line | p. 217 |
| Koenderink's theorems | p. 221 |
| Subparabolic lines | p. 223 |
| Uses for inversion | p. 229 |
| Exercises | p. 230 |
| Involutes of geodesic foliations | p. 233 |
| Introduction | p. 233 |
| Cuspidal edges | p. 234 |
| The involutes of a geodesic foliation | p. 240 |
| Coxeter groups | p. 248 |
| Exercises | p. 252 |
| The circles of a surface | p. 253 |
| Introduction | p. 253 |
| The theorems of Euler and Meusnier | p. 253 |
| Osculating circles | p. 255 |
| Contours and umbilical hill-tops | p. 260 |
| Higher order osculating circles | p. 263 |
| Exercises | p. 263 |
| Examples of surfaces | p. 265 |
| Introduction | p. 265 |
| Tubes | p. 265 |
| Ellipsoids | p. 266 |
| Symmetrical singularities | p. 270 |
| Bumpy spheres | p. 271 |
| The minimal monkey-saddle | p. 280 |
| Exercises | p. 285 |
| Flexcords of surfaces | p. 286 |
| Introduction | p. 286 |
| Umbilics of quadrics | p. 287 |
| Characterisations of flexcords | p. 288 |
| Birth of umbilics | p. 290 |
| Bumpy spheres | p. 298 |
| Exercises | p. 301 |
| Duality | p. 302 |
| Introduction | p. 302 |
| Curves in S[superscript 2] | p. 303 |
| Surfaces in S[superscript 3] | p. 306 |
| Curves in S[superscript 3] | p. 313 |
| Exercises | p. 316 |
| Further reading | p. 317 |
| References | p. 320 |
| Index | p. 327 |
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