Interpolating Cubic Splines,New

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 inSpace. 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 2DFunction Interpolation. 5.1 Lagrange Interpolating Polynomials. 5.2 Whittaker's Interpolation Formula. 5.3 Cubic Splines for 2DFunction 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 Bzier Polynomials. 8 Double Tangent Cubic Splines. 8.1 KochanekBartels Tangents. 8.2 FletcherMcAllister 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 vSplines. 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 BSpline 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 MultiPatch 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 3DFunction 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.