An Introduction to Splines for Use in Computer Graphics and Geometric Modeling
- 1st Edition - September 1, 1995
- Latest edition
- Authors: Richard H. Bartels, John C. Beatty, Brian A. Barsky
- Language: English
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
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Description
Description
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.
Table of contents
Table of contents
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
References
Index
Product details
Product details
- Edition: 1
- Latest edition
- Published: September 1, 1995
- Language: English