Interpolating Cubic Splines

Front Cover
Springer Science & Business Media, 2000 - Computers - 244 pages
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The study of spline functions is an outgrowth of basic mathematical concepts arising from calculus, analysis and numerical analysis. Spline modelling affects a number of fields: statistics; computer graphics; CAD programming, and other areas of applied mathematics.
 

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Contents

Mathematical Preliminaries
1
12 Vectors
3
13 Subspaces and Linear Independence
6
14 Vector Space Bases
8
15 Euclidean Length
11
16 The Euclidean Inner Product
12
17 Projection onto a Line
16
18 Planes in 3Space
20
82 FletcherMcAllister Tangent Magnitudes
97
Global Cubic Space Curve Splines
101
91 Second Derivatives of Global Cubic Splines
108
92 Third Derivatives of Global Cubic Splines
112
93 A Variational Characterization of Natural Splines
114
94 Weighted vSplines
116
Smoothing Splines
123
101 Computing an Optimal Smoothing Spline
124

19 Coordinate System Orientation
24
110 The Cross Product
26
Curves
31
21 The Tangent Curve
32
22 Curve Parameterization
34
23 The Normal Curve
36
24 Envelope Curves
37
25 Arc Length Parameterization
38
26 Curvature
39
27 The Frenet Equations
41
28 Involutes and Evolutes
43
29 Helices
45
210 Signed Curvature
46
211 Inflection Points
47
Surfaces
51
31 The Gradient of a Function
52
32 The Tangent Space and Normal Vector
54
33 Derivatives
55
Function and Space Curve Interpolation
59
2DFunction Interpolation
63
52 Whittakers Interpolation Formula
65
54 Estimating Slopes
68
55 Monotone 2D Cubic Spline Functions
69
56 Error in 2D Cubic Spline Interpolation Functions
72
ASpline Curves With Range Dimension d
75
Cubic Polynomial Space Curve Splines
77
71 Choosing the Segment Parameter Limits
81
72 Estimating Tangent Vectors
85
73 Bezier Polynomials
90
Double Tangent Cubic Splines
95
81 KochanekBartels Tangents
96
102 Computing the Smoothing Parameter
127
103 Best Fit Smoothing Cubic Splines
129
104 Monotone Smoothing Splines
130
Geometrically Continuous Cubic Splines
133
111 Beta Splines
136
Quadratic Space Curve Based Cubic Splines
139
Cubic Spline Vector Space Basis Functions
143
131 Bases for C and C Space Curve Cubic Splines
144
132 Cardinal Bases for Cubic Spline Vector Spaces
148
133 The BSpline Basis for Global Cubic Splines
151
Rational Cubic Splines
157
Two Spline Programs
159
152 Optimal Smoothing Spline Program
178
Tensor Product Surface Splines
193
162 A Generalized Tensor Product Patch Spline
197
163 Regular Grid MultiPatch Surface Interpolation
199
164 Estimating Tangent and Twist Vectors
200
165 Tensor Product Cardinal Basis Representation
203
166 Bicubic Splines with Variable Parameter Limits
205
168 Parametric Grids
207
169 3DFunction Interpolation
208
Boundary Curve Based Surface Splines
211
172 Boundary Curve Based Bicubic Interpolation
213
173 General Boundary Curve Based Spline Interpolation
215
Physical Splines
217
181 Computing a Space Curve Physical Spline Segment
222
182 Computing a 2D Physical Spline Segment
230
References
233
Index
237
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