Interpolating Cubic Splines

Springer Science & Business Media, 2000 - Computers - 244 pages
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 C¹ and C² 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 Copyright