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9780849394041

Handbook on Splines for the User

by ;
  • ISBN13:

    9780849394041

  • ISBN10:

    084939404X

  • Edition: Disk
  • Format: Hardcover
  • Copyright: 1995-07-14
  • Publisher: CRC Press
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List Price: $170.00

Summary

Splines find ever increasing application in the numerical methods, computer-aided design, and computer graphics areas. The Handbook on Splines for the User not only provides an excellent introduction to basic concepts and methods but also includes the SplineGuide-a computer diskette that allows the reader to practice using important programs.These programs help the user to build interpolating and smoothing cubic and bicubic splines of all classes. Programs are described in Fortran for spline functions and C for geometric splines.The Handbook describes spline functions and geometric splines and provides simple, but effective algorithms. It covers virtually all of the important types of cubic and bicubic splines, functions, variables, curves, and surfaces. The book is written in a straightforward manner and requires little mathematical background. When necessary, the authors give theoretical treatments in an easy-to-use form. Through the Handbook on Splines for the User, introduce yourself to the exciting world of splines and learn to use them in practical applications and computer graphics.

Table of Contents

Preface vii
Contents ix
Why splines? 1(4)
On the structure of the handbook 5(4)
Part 1. Spline functions 9(64)
Spline functions of one variable
11(36)
Interpolating cubic splines
13(19)
Statement of the interpolation problem
13(1)
Definition of the interpolating cubic spline function
14(1)
End (boundary) conditions
14(1)
Construction of the interpolating cubic spline function
15(2)
Advice for users
17(1)
Choice of the end conditions
17(1)
Choice of the interpolation knots
18(1)
Choice of the interpolating functions (pluses and minuses)
18(1)
The Lagrange interpolating polynomial
18(4)
Piecewise-linear interpolation
22(1)
Spline interpolation
23(2)
Properties of the interpolating cubic spline function
25(1)
Approximate properties
25(1)
Extreme property
26(1)
Construction of the interpolating spline curves with the help of the interpolating spline functions
26(4)
Implementation
30(2)
Smoothing cubic splines
32(10)
Statement of the smoothing problem
32(1)
Definition of the smoothing cubic spline function
33(1)
End (boundary) conditions
33(1)
Construction of the smoothing cubic spline function
34(2)
Choice of the weight coefficients
36(1)
Construction of the smoothing spline curves with the help of the smoothing spline functions
37(1)
Implementation
38(4)
Other types of splines
42(5)
Linear space of cubic spline functions
42(1)
Cubic B-spline functions
42(5)
Spline functions of two variables
47(26)
Interpolating bicubic splines
51(12)
Statement of the interpolating problem
51(1)
Definition of the interpolating bicubic spline function
51(1)
Boundary conditions
52(2)
Construction of the interpolating bicubic spline function
54(3)
Properties of the interpolating bicubic spline functions
57(1)
Approximate property
57(1)
Extreme property
57(1)
Construction of the interpolating spline surfaces with the help of the interpolating spline functions
58(2)
Implementation
60(3)
Smoothing bicubic splines
63(10)
Statement of the smoothing problem
63(1)
Definition of the smoothing bicubic spline function
63(1)
Boundary conditions
64(1)
Construction of the smoothing bicubic spline function
65(2)
Construction of the smoothing spline surfaces with the help of the smoothing spline functions
67(1)
Implementation
68(5)
Part II. Geometric splines 73(132)
Spline curves
75(66)
Main facts of the curve theory
78(13)
Parametrized curves
78(1)
Smooth and regular curves
78(1)
Transformation of parameter
79(1)
The Frenet trihedral
80(1)
Curvature and torsion of the curve
81(2)
Planar curves
83(1)
Parametric description
83(1)
Implicit description
84(1)
Composite curves
84(6)
Geometric continuity
90(1)
Bezier curves
91(13)
Parametric equations of an elementary Bezier curve
91(1)
Properties of the elementary Bezier curves
92(4)
Composite Bezier curves
96(3)
Rational Bezier curves
99(3)
Implementation
102(2)
B-spline curves
104(15)
Parametric equations of an elementary cubic B-spline curve
104(1)
Composite cubic B-spline curves
105(4)
Multiple and fantom vertices
109(5)
Rational cubic B-spline curves
114(1)
The Bezier form of the composite cubic B-spline curve
115(1)
Implementation
116(3)
Beta-spline curves
119(8)
Parametric equations of an elementary Beta-spline curve
119(2)
Composite Beta-spline curves
121(2)
Multiple and fantom vertices
123(3)
Implementation
126(1)
Other spline curves
127(14)
Interpolating cubic Hermite curves
127(6)
Spline curves of Catmull-Rom
133(4)
Composite planar implicit cubic curves
137(4)
Spline surfaces
141(64)
Main facts from the surface theory
145(14)
Parametrized surfaces
145(2)
Smooth and regular surfaces
147(1)
First fundamental form of a surface
147(2)
Curve on a surface
149(1)
Angle between curves on a surface
149(1)
Area of a surface
150(1)
Second fundamental form of a surface
151(1)
Lines of curvature
151(1)
Gaussian and mean curvatures
152(1)
Geometric continuity
152(4)
Twist vector and bilinear surface
156(3)
Bezier surfaces
159(10)
Parametric equations of an elementary Bezier surface
159(2)
Properties of the elementary Bezier surfaces
161(2)
Composite Bezier surfaces
163(2)
Rational Bezier surfaces
165(2)
Implementation
167(2)
B-spline surfaces
169(14)
Parametric equations of an elementary bicubic B-spline surface
169(2)
Properties of the elementary bicubic B-spline surfaces
171(1)
Composite bicubic B-spline surfaces
172(2)
Multiple and fantom vertices
174(6)
Rational bicubic B-spline surfaces
180(1)
Implementation
181(2)
Beta-spline surfaces
183(11)
Parametric equations of an elementary Beta-spline surface
183(2)
Properties of the elementary Beta-spline surfaces
185(1)
Composite Beta-spline surfaces
186(2)
Multiple and fantom vertices
188(4)
Implementation
192(2)
Other spline surfaces
194(11)
Interpolating bicubic Hermite surfaces
194(6)
Implementation
200(3)
Implicit cubic spline surfaces
203(2)
Appendix A. Programs of the sweep method for tridiagonal and pentadiagonal matrices 205(4)
Appendix B. Description of diskette 209(8)
Bibliography 217(2)
Index 219

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