Interpolating Cubic Splines

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.

1 Mathematical Preliminaries.- 1.1 The Pythagorean Theorem.- 1.2 Vectors.- 1.3 Subspaces and Linear Independence.- 1.4 Vector Space Bases.- 1.5 Euclidean Length.- 1.6 The Euclidean Inner Product.- 1.7 Projection onto a Line.- 1.8 Planes in-Space.- 1.9 Coordinate System Orientation.- 1.10 The Cross Product.- 2 Curves.- 2.1 The Tangent Curve.- 2.2 Curve Parameterization.- 2.3 The Normal Curve.- 2.4 Envelope Curves.- 2.5 Arc Length Parameterization.- 2.6 Curvature.- 2.7 The Frenet Equations.- 2.8 Involutes and Evolutes.- 2.9 Helices.- 2.10 Signed Curvature.- 2.11 Inflection Points.- 3 Surfaces.- 3.1 The Gradient of a Function.- 3.2 The Tangent Space and Normal Vector.- 3.3 Derivatives.- 4 Function and Space Curve Interpolation.- 5 2D-Function Interpolation.- 5.1 Lagrange Interpolating Polynomials.- 5.2 Whittaker''s Interpolation Formula.- 5.3 Cubic Splines for 2D-Function Interpolation.- 5.4 Estimating Slopes.- 5.5 Monotone 2D Cubic Spline Functions.- 5.6 Error in 2D Cubic Spline Interpolation Functions.- 6 ?-Spline Curves With Range Dimension d.- 7 Cubic Polynomial Space Curve Splines.- 7.1 Choosing the Segment Parameter Limits.- 7.2 Estimating Tangent Vectors.- 7.3 Bézier Polynomials.- 8 Double Tangent Cubic Splines.- 8.1 Kochanek-Bartels Tangents.- 8.2 Fletcher-McAllister Tangent Magnitudes.- 9 Global Cubic Space Curve Splines.- 9.1 Second Derivatives of Global Cubic Splines.- 9.2 Third Derivatives of Global Cubic Splines.- 9.3 A Variational Characterization of Natural Splines.- 9.4 Weighted v-Splines.- 10 Smoothing Splines.- 10.1 Computing an Optimal Smoothing Spline.- 10.2 Computing the Smoothing Parameter.- 10.3 Best Fit Smoothing Cubic Splines.- 10.4 Monotone Smoothing Splines.- 11 Geometrically Continuous Cubic Splines.- 11.1 Beta Splines.- 12 Quadratic Space Curve Based Cubic Splines.- 13 Cubic Spline Vector Space Basis Functions.- 13.1 Bases for C1 and C2 Space Curve Cubic Splines.- 13.2 Cardinal Bases for Cubic Spline Vector Spaces.- 13.3 The B-Spline Basis for Global Cubic Splines.- 14 Rational Cubic Splines.- 15 Two Spline Programs.- 15.1 Interpolating Cubic Splines Program.- 15.2 Optimal Smoothing Spline Program.- 16 Tensor Product Surface Splines.- 16.1 Bicubic Tensor Product Surface Patch Splines.- 16.2 A Generalized Tensor Product Patch Spline.- 16.3 Regular Grid Multi-Patch Surface Interpolation.- 16.4 Estimating Tangent and Twist Vectors.- 16.5 Tensor Product Cardinal Basis Representation.- 16.6 Bicubic Splines with Variable Parameter Limits.- 16.7 Triangular Patches.- 16.8 Parametric Grids.- 16.9 3D-Function Interpolation.- 17 Boundary Curve Based Surface Splines.- 17.1 Boundary Curve Based Bilinear Interpolation.- 17.2 Boundary Curve Based Bicubic Interpolation.- 17.3 General Boundary Curve Based Spline Interpolation.- 18 Physical Splines.- 18.1 Computing a Space Curve Physical Spline Segment.- 18.2 Computing a 2D Physical Spline Segment.- References.