Handbook on Splines for the User

by Shikin; Eugene V.

• ISBN13: 9780849394041
• ISBN10: 084939404X
• Edition: Disk
• Format: Hardcover
• Copyright: 1995-07-14
• Publisher: CRC Press

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