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Curves and surfaces in computer aided geometric design - Yamaguchi F.

Yamaguchi F. Curves and surfaces in computer aided geometric design - Tokyo, 1988. - 390 p.
Download (direct link): curvesandsurfacesincomputer1988.djvu
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References...................................................................... 134
4. Spline Interpolation....................................................... 135
4.1 Splines.................................................................... 135
4.2 Spline Functions........................................................... 136
4.3 Mathematical Representation of Spline Functions............................ 137
4.4 Natural Splines............................................................ 138
4.5 Natural Splines and the Minimum Interpolation Property..................... 140
4.6 Smoothing Splines.......................................................... 144
4.7 Parametric Spline Curves................................................... 145
Contents
IX
4.8 End Conditions on a Spline Curve......................................... 151
4.9 Cubic Spline Curves Using Circular Arc Length............................ 159
4.10 B-Splines................................................................ 160
4.11 Generation of Spline Surfaces............................................ 163
References.................................................................... 168
5. The Bernstein Approximation ............................................. 169
5.1 Curves................................................................... 169
5.1.1 Modification of Ferguson Curve Segments........................... 169
5.1.2 Cubic Bezier Curve Segments....................................... 173
5.1.3 Bezier Curve Segments............................................. 182
5.1.4 Properties of the Bernstein Basis Function and
Bernstein Polynomial............................................. 189
5.1.5 Various Representations for Bezier Curve Segments................. 193
5.1.6 Derivative Vectors of Bezier Curve Segments....................... 198
5.1.7 Determination of a Point on a Curve Segment
by Linear Operations............................................. 199
5.1.8 Increase of the Degree of a Bezier Curve Segment..................204
5.1.9 Partitioning of a Bezier Curve Segment............................209
5.1.10 Connection of Bezier Curve Segments...............................213
5.1.11 Creation of a Spline Curve with Cubic Bezier Curve Segments 214
5.2 Surfaces.................................................................216
5.2.1 Bezier Surface Patches ...........................................216
5.2.2 The Relation Between a Bi-cubic Bezier Surface Patch
and a Bi-cubic Coons Surface Patch...............................218
5.2.3 Connection of Bezier Surface Patches .............................221
5.2.4 Triangular Patches Formed by Degeneration.........................226
5.2.5 Triangular Patches ...............................................228
5.2.6 Some Considerations on Bezier Curves and Surfaces.................231
References....................................................................232
6. The -Spline Approximation...............................................233
6.1 Uniform Cubic -Spline Curves............................................233
6.1.1 Derivation of the Curve Formula...................................233
6.1.2 Properties of Curves..............................................240
6.1.3 Determination of a Point on a Curve by Finite Difference
Operations.......................................................246
6.1.4 Inverse Transformation of a Curve.................................247
6.1.5 Change of Polygon Vertices........................................250
6.2 Uniform Bi-cubic -Spline Surfaces ......................................251
6.2.1 Surface Patch Formulas............................................251
6.2.2 Determination of a Point on a Surface by Finite Difference
Operations.......................................................259
6.2.3 Inverse Transformation of a Surface...............................262
6.2.4 Surfaces of Revolution............................................265
X
Contents
6.3 -Spline Functions and Their Properties (1) ...........................270
6.4 -Spline Functions and Their Properties (2) ...........................272
6.5 Derivation of -Spline Functions..........................................274
6.6 -Spline Curve Type (1) ..................................................281
6.7 -Spline Curve Type (2) ..................................................283
6.8 Recursive Calculation of -Spline Functions...............................285
6.9 -Spline Functions and Their Properties (3) ...........................291
6.10 -Spline Curve Type (3) ..................................................294
6.11 Differentiation of -Spline Curves........................................309
6.12 Geometrical Properties of -Spline Curves ................................ 311
6.13 Determination of a Point on a Curve by Linear Operations..................312
6.14 Insertion of Knots ....................................................... 316
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