Geometric continuity of curves and surfaces

Contents......Page 11
Preface......Page 15
Notation......Page 17
Introduction......Page 19
 Equations of geometric continuity......Page 25
 Interpretation......Page 29
  -spline curves......Page 32
  -spline curves......Page 34
  -spline curves......Page 50
 Tensor product geometric spline patches......Page 54
 Geometric continuity at a common boundary......Page 57
 Interpretation......Page 60
 A little bit of algebra......Page 62
 Polynomial solutions of equations  of geometric continuity......Page 65
 Constructing pairs of patches......Page 71
 Approximating smooth junctions......Page 75
  Joining patches along constant parameter curves......Page 76
  Joining a spline patch to a trimmed spline patch......Page 79
 Hahn's scheme of filling polygonal holes......Page 87
 Compatibility conditions at a common corner......Page 89
  G1 compatibility conditions......Page 93
  Piecewise polynomial functions in R2......Page 98
  Connection with trigonometric splines......Page 106
  Trigonometric splines and G1  compatibility conditions......Page 109
  Trigonometric splines and  G2 compatibility conditions......Page 112
 Beyond the curvature continuity and towards practice......Page 126
Filling polygonal holes......Page 127
 Theoretical background......Page 128
 Constructing function spaces......Page 132
 Minimisation of quadratic forms......Page 147
 Constructions with shape optimisation......Page 155
 Conclusion......Page 159
Images of surface shape......Page 163
 Characteristic lines and shape functions......Page 164
 Planar sections......Page 166
 Isophotes......Page 167
 Reflection lines......Page 168
 Highlight lines......Page 172
 Surface curvatures......Page 173
 Lagrange and Hermite interpolation......Page 177
  The divided differences algorithm......Page 178
  Definition......Page 181
  Derivatives......Page 182
  Algebraic operations......Page 184
 Bézier patches......Page 187
 B-spline curves......Page 189
  Normalised B-spline functions......Page 190
  Nonuniform B-spline curves......Page 197
  B-spline curves with uniform knots......Page 201
 Tensor product B-spline patches......Page 204
  B-spline patches with uniform knots......Page 206
  Mesh representation of surfaces......Page 207
  Mesh refinement......Page 208
  Extraordinary elements of meshes......Page 210
  The limiting surface......Page 211
 Rational curves and patches......Page 215
  Blending polynomials......Page 217
  Cubic splines of interpolation—Hermite form......Page 219
  Cubic B-splines of interpolation......Page 221
  Approximation properties......Page 222
  Bilinearly blended Coons patches......Page 226
  Bicubically and  biquintically blended Coons patches......Page 227
  Coons patches in Bézier form......Page 230
  Curvatures of curves......Page 231
  Curvatures of surfaces......Page 233
 Fàa di Bruno's formula......Page 236
Bibliography......Page 239
Author's Biography......Page 245
Index......Page 247