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An Introduction to Splines for Use in Computer Graphics and Geometric Modeling - 1st Edition - ISBN: 9781558604001, 9780080509211

An Introduction to Splines for Use in Computer Graphics and Geometric Modeling

1st Edition

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Authors: Richard Bartels John Beatty Brian Barsky
Paperback ISBN: 9781558604001
eBook ISBN: 9780080509211
Imprint: Morgan Kaufmann
Published Date: 17th April 1996
Page Count: 476
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Table of Contents

An Introduction to Splines for Use in Computer Graphics and Geometric Modeling
by Richard H. Bartels, John C. Beatty, and Brian A. Barsky

    1 Introduction
      1.1 General References

    2 Preliminaries
    3 Hermite and Cubic Spline Interpolation
      3.1 Practical Considerations - Computing Natural Cubic Splines
      3.2 Other End Conditions For Cubic Interpolating Splines
      3.3 Knot Spacing
      3.4 Closed Curves

    4 A Simple Approximation Technique - Uniform Cubic B-splines
      4.1 Simple Preliminaries - Linear B-splines
      4.2 Uniform Cubic B-splines
      4.3 The Convex Hull Property
      4.4 Translation Invariance
      4.5 Rotation and Scaling Invariance
      4.6 End Conditions for Curves
      4.7 Uniform Bicubic B-spline Surfaces
      4.8 Continuity for Surfaces
      4.9 How Many Patches Are There?
      4.10 Other Properties
      4.11 Boundary Conditions for Surfaces

    5 Splines in a More General Setting
      5.1 Preliminaries
      5.2 Continuity
      5.3 Segment Transitions
      5.4 Polynomials
      5.5 Vector Spaces
      5.6 Polynomials as a Vector Space
      5.7 Bases and Dimension
      5.8 Change of Basis
      5.9 Subspaces
      5.10 Knots and Parameter Ranges: Splines as a Vector Space
      5.11 Spline Continuity and Multiple Knots

    6 The One-Sided Basis
      6.1 The One-Sided Cubic
      6.2 The General Case
      6.3 One-Sided Basis
      6.5 Linear Combinations and Cancellation
      6.6 Cancellation as a Divided Difference
      6.7 Cancelling the Quadratic Term - The Second Difference
      6.8 Cancelling the Linear Term - The Third Difference
      6.9 The Uniform Cubic B-Spline - A Fourth Difference

    7 Divided Differences
      7.1 Differentiation and One-Sided Power Functions
      7.2 Divided Differences in a General Setting
      7.3 Algebraic and Analytic Properties

    8 General B-splines
      8.1 A Simple Example - Step Function B-splines
      8.2 Linear B-splines
      8.3 General B-spline Bases
      8.4 Examples - Quadratic B-splines
      8.5 The Visual Effect of Knot Multiplicities - Cubic B-splines
      8.6 Altering Knot Spacing - More Cubic B-splines

    9 B-spline Properties
      9.1 Differencing Products - The Leibniz Rule
      9.2 Establishing a Recurrence
      9.3 The Recurrence and Examples
      9.4 Evaluating B-splines Through Recurrence
      9.5 Compact Support, Positivity, and the Convex Hull Property
      9.6 Practical Implications

    10 Bezier Curves
      10.1 Increasing the Degree of a Bezier Curve
      10.2 Composite Bezier Curves
      10.3 Local vs. Global Curves
      10.4 Subdivision and Refinement
      10.5 Midpoint Subdivision of Bezier Curves
      10.6 Arbitrary Subdivision of Bezier Curves
      10.7 Bezier Curves From B-Splines
      10.8 A Matrix Formulation
      10.9 Converting Between Representations
      10.10 Bezier Surfaces

    11. Knot Insertion
      11.1 Knots and Vertices
      11.2 Representation Results

    12 The Oslo Algorithm
      12.1 Discrete B-spline Recurrence
      12.2 Discrete B-spline Properties
      12.3 Control Vertex Recurrence
      12.4 Illustrations

    13 Parametric vs. Geometric Continuity
      13.1 Geometric Continuity
      13.2 Continuity of the First Derivative Vector
      13.3 Continuity of the Second Derivative Vector

    14 Uniformly-Shaped Beta-spline Surfaces
      14.1 Uniformly-Shaped Beta-spline Surfaces
      14.2 An Historical Note

    15 Geometric Continuity, Reparametrization, and the Chain Rule
    16 Continuously-Shaped Beta-splines
      16.1 Locality
      16.2 Bias
      16.3 Tension
      16.4 Convex Hull
      16.5 End Conditions
      16.6 Evaluation
      16.7 Continuously-Shaped Beta-spline Surfaces

    17 An Explicity Formulation for Cubic Beta-splines
      17.1 Beta-splines with Uniform Knot Spacing
      17.2 Formulas
      17.3 Recurrence
      17.4 Examples

    18 Discretely-Shaped Beta-splines
      18.1 A Truncated Power Basis for the Beta-splines
      18.2 A Local Basis for the Beta-splines
      18.3 Evaluation
      18.4 Equivalence
      18.5 Beta2-splines
      18.6 Examples

    19 B-spline Representations for Beta-splines
      19.1 Linear Equations
      19.2 Examples

    20 Rendering and Evaluation
      20.1 Values of B-splines
      20.2 Sums of B-splines
      20.3 Derivatives of B-splines
      20.4 Conversion to Segment Polynomials
      20.5 Rendering Curves: Horner's Rule and Forward Differencing
      20.6 The Oslo Algorithm - Computing Discrete B-splines
      20.7 Parial Derivatives and Normals
      20.8 Locality
      20.9 Scan-Line Methods
      20.10 Ray-Tracing B-spline Surfaces

    21 Selected Applications
      21.1 The Hermite Basis and C1 Key-Frame Inbetweening
      21.2 A Cardinal Basis Spline for Interpolation
      21.3 Interpolation Using B-splines
      21.4 Catmull-Rom Splines
      21.5 B-splines and Least Squares Fitting


As the field of computer graphics develops, techniques for modeling complex curves and surfaces are increasingly important. A major technique is the use of parametric splines in which a curve is defined by piecing together a succession of curve segments, and surfaces are defined by stitching together a mosaic of surface patches.

An Introduction to Splines for Use in Computer Graphics and Geometric Modeling discusses the use of splines from the point of view of the computer scientist. Assuming only a background in beginning calculus, the authors present the material using many examples and illustrations with the goal of building the reader's intuition. Based on courses given at the University of California, Berkeley, and the University of Waterloo, as well as numerous ACM Siggraph tutorials, the book includes the most recent advances in computer-aided geometric modeling and design to make spline modeling techniques generally accessible to the computer graphics and geometric modeling communities.


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© Morgan Kaufmann 1987
17th April 1996
Morgan Kaufmann
Paperback ISBN:
eBook ISBN:

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About the Authors

Richard Bartels

John Beatty

Brian Barsky