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Interpolating Cubic Splines

  • Book
  • © 2000

Overview

Authors:
  1. Gary D. Knott

    1. Civilized Software, Inc., Silver Spring, USA

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Part of the book series: Progress in Computer Science and Applied Logic (PCS, volume 18)

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Table of contents (18 chapters)

  1. Front Matter

    Pages i-xii
    Download chapter PDF

  2. Mathematical Preliminaries

    • Gary D. Knott
    Pages 1-29

  3. Curves

    • Gary D. Knott
    Pages 31-49

  4. Surfaces

    • Gary D. Knott
    Pages 51-57

  5. Function and Space Curve Interpolation

    • Gary D. Knott
    Pages 59-61

  6. 2D-Function Interpolation

    • Gary D. Knott
    Pages 63-73

  7. Λ-Spline Curves With Range Dimension d

    • Gary D. Knott
    Pages 75-76

  8. Cubic Polynomial Space Curve Splines

    • Gary D. Knott
    Pages 77-93

  9. Double Tangent Cubic Splines

    • Gary D. Knott
    Pages 95-100

  10. Global Cubic Space Curve Splines

    • Gary D. Knott
    Pages 101-121

  11. Smoothing Splines

    • Gary D. Knott
    Pages 123-132

  12. Geometrically Continuous Cubic Splines

    • Gary D. Knott
    Pages 133-138

  13. Quadratic Space Curve Based Cubic Splines

    • Gary D. Knott
    Pages 139-142

  14. Cubic Spline Vector Space Basis Functions

    • Gary D. Knott
    Pages 143-155

  15. Rational Cubic Splines

    • Gary D. Knott
    Pages 157-158

  16. Two Spline Programs

    • Gary D. Knott
    Pages 159-191

  17. Tensor Product Surface Splines

    • Gary D. Knott
    Pages 193-209

  18. Boundary Curve Based Surface Splines

    • Gary D. Knott
    Pages 211-216

  19. Physical Splines

    • Gary D. Knott
    Pages 217-232

  20. Back Matter

    Pages 233-244
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Keywords

About this book

A spline is a thin flexible strip composed of a material such as bamboo or steel that can be bent to pass through or near given points in the plane, or in 3-space in a smooth manner. Mechanical engineers and drafting specialists find such (physical) splines useful in designing and in drawing plans for a wide variety of objects, such as for hulls of boats or for the bodies of automobiles where smooth curves need to be specified. These days, physi­ cal splines are largely replaced by computer software that can compute the desired curves (with appropriate encouragment). The same mathematical ideas used for computing "spline" curves can be extended to allow us to compute "spline" surfaces. The application ofthese mathematical ideas is rather widespread. Spline functions are central to computer graphics disciplines. Spline curves and surfaces are used in computer graphics renderings for both real and imagi­ nary objects. Computer-aided-design (CAD) systems depend on algorithms for computing spline functions, and splines are used in numerical analysis and statistics. Thus the construction of movies and computer games trav­ els side-by-side with the art of automobile design, sail construction, and architecture; and statisticians and applied mathematicians use splines as everyday computational tools, often divorced from graphic images.

Reviews

"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."

--Zentralblatt Math

Authors and Affiliations

  • Civilized Software, Inc., Silver Spring, USA

    Gary D. Knott

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