# Interpolating Cubic Splines

Birkhauser, 2000 - Computers - 244 pages
To download the source files for the two spline programs, please scroll to chapter 15 in the below table-of-contents. For additional information, visit the http://www.civilized.com author's website. Spline functions arise in a number of fields: statistics, computer graphics, programming, computer-aided design technology, numerical analysis, and other areas of applied mathematics. Much work has focused on approximating splines such as B-splines and Bezier splines. In contrast, this book emphasizes interpolating splines. Almost always, the cubic polynomial form is treated in depth. Interpolating Cubic Splines covers a wide variety of explicit approaches to designing splines for the interpolation of points in the plane by curves, and the interpolation of points in 3-space by surfaces. These splines include various estimated-tangent Hermite splines and double-tangent splines, as well as classical natural splines and geometrically-continuous splines such as beta-splines and n-splines. A variety of special topics are covered, including monotonic splines, optimal smoothing splines, basis representations, and exact energy-minimizing physical splines. An in-depth review of the differential geometry of curves and a broad range of exercises, with selected solutions, and complete computer programs for several forms of splines and smoothing splines, make this book useful for a broad audience: students, applied mathematicians, statisticians, engineers, and practicing programmers involved in software development in computer graphics, CAD, and various engineering applications. Series: Progress in Computer Science, Vol. 18 Contents Preface 1. Mathematical Preliminaries 1.1 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 3-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 of a Surface 3.3 Derivatives 4. Function and Space Curve Interpolation 5. 2-D Function Interpolation 5.1 Lagrange Interpolating Polynomials 5.2 Whittaker's Interpolation Formula 5.3 Cubic Polynomial Splines for 2D-Function Interpolation 5.4 Estimating Slopes 5.5 Monotone 2D Cubic Spline Interpolation 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 Bezier Polynomials 8. Double-Tangent Cubic Splines 8.1 Kochanek-Bartels Tangents 8.2 Knuth Tangent Magnitudes 8.3 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 C 1 and C 2 Space Curve Cubic Splines 13.2 Cardinal Bases for Vector Spaces of Cubic Splines 13.3 The B-Spline Basis for Global Cubic Splines 14. Rational Cubic Splines 15. Two Spline Programs 15.1 Interpolating Cubic Splines Program(Click http://www.birkhauser.com/Books/ISBN/0-8176-4100-9/gspline.c to view the source code, and use your browser to save it locally.) 15.2 Optimal Smoothing Spline Program(Click http://www.birkhauser.com/Books/ISBN/0-8176-4100-9/ssp.c to view the source code, and use your browser to save it locally.) 16. Tensor Product Surface Splines 16.1 Bicubic Tensor Product Surface Patch Splines 16.2 A Generalized Tensor Product Patch Interpolation Spline 16.3 Regular-Parameter-Grid Multi-Patch Surface Interpolation 16.4 Estimating Tangent and Twist Vectors 16.5 Tensor Product Cardinal Basis Representation 16.6 Extended 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 Physical Spline Connecting Two Points References Index

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