Interpolating Cubic Splines: 18 (Progress In Computer Science And Applied Logic

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 B?zier Polynomials.- 8 Double Tangent Cubic Splines.- 8.1 Kochanek-Bartels Tangents.- 8.2 Fletcher-McAllister Tangent Magnitudes.- 9 Global Cubic Space Curve Splines.