<P>Spline functions arise in a number of fields: statistics, computer graphics, programming, computer-aided design technology, numerical analysis, and other areas of applied mathematics. </P> <P>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. </P> <P><I>Interpolating Cubic Splines</I> 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.  </P> <P>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. </P>  <P><B>Series: </B>Progress in Computer Science, Vol 18</P>  <HR WIDTH="100%">  <P> <B>Contents</B> </P>  <P> <B>Preface</B> </P> <P> <B>1. Mathematical Preliminaries</B> <UL>1.1 Pythagorean Theorem<BR> 1.2 Vectors<BR> 1.3 Subspaces and Linear Independence<BR> 1.4 Vector Space Bases<BR> 1.5 Euclidean Length<BR> 1.6 The Euclidean Inner Product<BR> 1.7 Projection onto a Line<BR> 1.8 Planes in 3-Space<BR> 1.9 Coordinate System Orientation<BR> 1.10 The Cross Product </UL></P> <P><B>2. Curves</B> <UL>2.1 The Tangent Curve<BR> 2.2 Curve Parameterization<BR> 2.3 The Normal Curve<BR> 2.4 Envelope Curves<BR> 2.5 Arc-Length Parameterization<BR> 2.6 Curvature<BR> 2.7 The Frenet Equations<BR> 2.8 Involutes and Evolutes<BR> 2.9 Helices<BR> 2.10 Signed Curvature<BR> 2.11 Inflection Points </UL></P> <P><B>3. Surfaces</B> <UL>3.1 The Gradient of a Function<BR> 3.2 The Tangent Space and Normal Vector of a Surface<BR> 3.3 Derivatives </UL></P> <P><B>4. Function and Space Curve Interpolation</B></P> <P><B>5. 2-D Function Interpolation</B> <UL>5.1 Lagrange Interpolating Polynomials<BR> 5.2 Whittaker's Interpolation Formula<BR> 5.3 Cubic Polynomial Splines for 2D-Function Interpolation<BR> 5.4 Estimating Slopes<BR> 5.5 Monotone 2D Cubic Spline Interpolation Functions<BR> 5.6 Error in 2D Cubic Spline Interpolation Functions </UL></P> <P><B>6. Spline Curves with Range-Dimension <I>d</I></B></P> <P><B>7. Cubic Polynomial Space Curve Splines</B> <UL>7.1 Choosing the Segment Parameter Limits<BR> 7.2 Estimating Tangent Vectors<BR> 7.3 Bezier Polynomials </UL></P> <P><B>8. Double-Tangent Cubic Splines</B> <UL>8.1 Kochanek-Bartels Tangents<BR> 8.2 Knuth Tangent Magnitudes<BR> 8.3 Fletcher-McAllister Tangent Magnitudes </UL></P> <P><B>9. Global Cubic Space Curve Splines</B> <UL>9.1 Second-Derivatives of Global Cubic Splines<BR> 9.2 Third-Derivatives of Global Cubic Splines<BR> 9.3 A Variational Characterization of Natural Splines<BR> 9.4 Weighted <I>v</I>-Splines </UL></P> <P><B>10. Smoothing Splines</B> <UL>10.1 Computing an Optimal Smoothing Spline<BR> 10.2 Computing the Smoothing Parameter<BR> 10.3 Best-Fit Smoothing Cubic Splines<BR> 10.4 Monotone Smoothing Splines </UL></P> <P><B>11. Geometrically-Continuous Cubic Splines</B> <UL>11.1 Beta Splines </UL></P> <P><B>12. Quadratic Space Curve-Based Cubic Splines</B></P> <P><B>13. Cubic Spline Vector Space Basis Functions</B> <UL>13.1 Bases for <I>C</I><SUP>1</SUP> and <I>C</I><SUP>2</SUP> Space Curve Cubic Splines<BR> 13.2 Cardinal Bases for Vector Spaces of Cubic Splines<BR> 13.3 The B-Spline Basis for Global Cubic Splines </UL></P> <P><B>14. Rational Cubic Splines</B></P> <P><B>15. Two Spline Programs</B> <UL>15.1 Interpolating Cubic Splines Program(<A HREF="http://www.birkhauser.com/Books/ISBN/0-8176-4100-9/source1.tex">Source</A>)<BR> 15.2 Optimal Smoothing Spline Program(<A HREF="http://www.birkhauser.com/Books/ISBN/0-8176-4100-9/source2.tex">Source</A>) </UL></P> <P><B>16. Tensor Product Surface Splines</B> <UL>16.1 Bicubic Tensor Product Surface Patch Splines<BR> 16.2 A Generalized Tensor Product Patch Interpolation Spline<BR> 16.3 Regular-Parameter-Grid Multi-Patch Surface Interpolation<BR> 16.4 Estimating Tangent and Twist Vectors<BR> 16.5 Tensor Product Cardinal Basis Representation<BR> 16.6 Extended Bicubic Splines with Variable Parameter Limits<BR> 16.7 Triangular Patches<BR> 16.8 Parametric Grids<BR> 16.9 3D-Function Interpolation </UL></P> <P><B>17. Boundary-Curve Based Surface Splines</B> <UL>17.1 Boundary-Curve-Based Bilinear Interpolation<BR> 17.2 Boundary-Curve-Based Bicubic Interpolation<BR> 17.3 General Boundary-Curve-Based Spline Interpolation </UL></P> <P><B>18. Physical Splines</B> <UL>18.1 Computing a Physical Spline Connecting Two Points </UL></P> <P><B> References</B></P>  <P><B>Index</B></P> 	
