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

## 1st Edition

**Authors:**Richard Bartels John Beatty Brian Barsky

**Paperback ISBN:**9781558604001

**eBook ISBN:**9780080509211

**Imprint:**Morgan Kaufmann

**Published Date:**1st September 1995

**Page Count:**476

**View all volumes in this series:**The Morgan Kaufmann Series in Computer Graphics

## 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

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

Index

## Description

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

by Richard H. Bartels, John C. Beatty, and Brian A. Barsky

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

Index

## Details

- No. of pages:
- 476

- Language:
- English

- Copyright:
- © Morgan Kaufmann 1987

- Published:
- 1st September 1995

- Imprint:
- Morgan Kaufmann

- eBook ISBN:
- 9780080509211

- Paperback ISBN:
- 9781558604001