LIMITED OFFER

## Save 50% on book bundles

Immediately download your ebook while waiting for your print delivery. No promo code is needed.

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

## Purchase options

## Save 50% on book bundles

## Institutional subscription on ScienceDirect

Request a sales quote**An Introduction to Splines for Use in Computer Graphics and Geometric Modeling**

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

Save up to 20% on Elsevier print and eBooks with free shipping. No promo code needed.

Save up to 20% on print and eBooks.

1st Edition - September 1, 1995

Authors: Richard H. Bartels, John C. Beatty, Brian A. Barsky

Language: EnglishPaperback ISBN:

9 7 8 - 1 - 5 5 8 6 0 - 4 0 0 - 1

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 wh… Read more

LIMITED OFFER

Immediately download your ebook while waiting for your print delivery. No promo code is needed.

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.

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

- 1.1 General References

- 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

- 4.1 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

- 5.1 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

- 6.1 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

- 7.1 Differentiation and One-Sided Power Functions

7.2 Divided Differences in a General Setting

7.3 Algebraic and Analytic Properties

- 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

- 9.1 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

- 10.1 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

- 11.1 Knots and Vertices

11.2 Representation Results

- 12.1 Discrete B-spline Recurrence

12.2 Discrete B-spline Properties

12.3 Control Vertex Recurrence

12.4 Illustrations

- 13.1 Geometric Continuity

13.2 Continuity of the First Derivative Vector

13.3 Continuity of the Second Derivative Vector

- 14.1 Uniformly-Shaped Beta-spline Surfaces

14.2 An Historical Note

- 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

- 17.1 Beta-splines with Uniform Knot Spacing

17.2 Formulas

17.3 Recurrence

17.4 Examples

- 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

- 19.1 Linear Equations

19.2 Examples

- 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

- 21.1 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

References

Index

- No. of pages: 476
- Language: English
- Edition: 1
- Published: September 1, 1995
- Imprint: Morgan Kaufmann
- Paperback ISBN: 9781558604001