Interpolating Cubic Splines

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 Whittakers 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 Bzier 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