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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.
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0.2 Processing and Analysis of Shapes ........................................ 2
0.3 Mathematical Description of Free Form Shapes.............................. 3
0.4 The Development of Mathematical Descriptions of Free Form Curves
and Surfaces............................................................. 4
References.................................................................... 10
1. Basic Theory of Curves and Surfaces...................................... 11
1.1 General.................................................................. 11
1.1.1 Properties of Object Shapes and Their Mathematical
Representation................................................... 11
1.1.2 Design and Mathematical Representations........................... 14
1.1.3 Invariance of a Shape Under Coordinate Transformation .... 18
1.2 Curve Theory............................................................. 20
1.2.1 Parametric Representation of Curves; Tangent Lines and
Osculating Planes................................................ 20
1.2.2 Curvature and Torsion............................................. 24
1.2.3 Frenet Frames and the Frenet-Serret Equations..................... 29
1.2.4 Calculation of a Point on a Curve................................. 31
1.2.5 Connection of Curve Segments...................................... 33
1.2.6 Parameter Transformation.......................................... 35
1.2.7 Partitioning of a Curve Segment................................... 39
1.2.8 Parametric Cubic Curves........................................... 39
1.2.9 Length and Area of a Curve ................................. 42
1.2.10 Intersection of a Curve with a Plane ............................. 43
1.2.11 Intersection of Two Curves........................................ 43
1.3 Theory of Surfaces....................................................... 44
1.3.1 Parametric Representation of Surfaces............................. 44
1.3.2 The First Fundamental Matrix of a Surface......................... 46
1.3.3 Determining Conditions for a Tangent Vector to a Curve
on a Surface..................................................... 48
1.3.4 Curvature of a Surface............................................ 49
1.3.5 Calculation of a Point on a Surface............................... 51
1.3.6 Subdivision of Surface Patches.................................... 55
1.3.7 Connection of Surface Patches .................................... 57
VIII Contents
1.3.8 Degeneration of a Surface Patch..................................... 59
1.3.9 Calculation of a Normal Vector on a Surface......................... 60
1.3.10 Calculation of Surface Area and Volume of a Surface............ 61
1.3.11 Offset Surfaces..................................................... 63
References...................................................................... 63
2. Lagrange Interpolation..................................................... 64
2.1 Lagrange Interpolation Curves ............................................. 64
2.2 Expression in Terms of Divided Differences................................. 67
References...................................................................... 71
3. Hermite Interpolation...................................................... 72
3.1 Hermite Interpolation...................................................... 72
3.2 Curves..................................................................... 73
3.2.1 Derivation of a Ferguson Curve Segment.............................. 73
3.2.2 Approximate Representation of a Circular Arc by a Ferguson
Curve Segment...................................................... 78
3.2.3 Hermite Interpolation Curves........................................ 80
3.2.4 Partitioning of Ferguson Curve Segments............................. 84
3.2.5 Increase of Degree of a Ferguson Curve Segment ..................... 85
3.3 Surfaces................................................................... 87
3.3.1 Ferguson Surface Patch.............................................. 87
3.3.2 The Coons Surface Patches (1964) ................................ 91
3.3.3 The Coons Surface Patches (1967) ................................ 102
3.3.4 Twist Vectors and Surface Shapes ................................... 112
3.3.5 Methods of Determining Twist Vectors................................ 117
3.3.6 Partial Surface Representation of the Coons Bi-cubic Surface
Patch.............................................................. 122
3.3.7 Connection of the Coons Bi-cubic Surface Patches.................... 123
3.3.8 Shape Control of the Coons Bi-cubic Surface Patch................... 126
3.3.9 Triangular Patches Formed by Degeneration......................... 131
3.3.10 Decomposition of Coons Surface Patches and 3 Types in
Constructing Surfaces.............................................. 132
3.3.11 Some Considerations on Hermite Interpolation Curves
and Surfaces....................................................... 133
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