1. Lee Byung-Gook Dongseo Univ. http://kowon.dongseo.ac.kr/~lbg/
Spline Methods in CAGD
Lee Byung-Gook
Dongseo Univ.
KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ.,

2. Affine combination Linear combinations
Affine(Barycentric) combinations
Convex combinations
Barycentric coordinates
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3. Affine combination Euclidean coordinate system Coordinate-free system
convex, barycentric combination t0<t<t1
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4. Polynomial interpolation
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5. Polynomial interpolation
Lagrange polynomials
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6. Examples of cubic interpolation
not convex hull property
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7. Bezier not interpolation convex hull property
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8. Representation Bezier
\sum_{i=0}^d b_{i,d}(t)=1 convex hull property
maximum at t=i/d
symmetric
\sum_{i=0}^d i/d b_{i,d}(t)=t linear precision
reproduce straight line
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9. Properties of Bezier Affine invariance Convex hull property
Endpoint interpolation
Symmetry
Linear precision
Pseudo-local control
Variation Diminishing Property
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10. Linear splines
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11. Quadratic splines
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12. Quadratic splines
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13. Representation splines
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14. B-spline Recurrence Relation Bernstein polynomial
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15. B-spline
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16. B-spline Smoothness=Degree-Multiplicity cubic case c2 c1 c0
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17. Spline space
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18. Univariate spline
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19. Cubic splines
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20. Bezier Paul de Faget de Casteljau, Citroen, 1959
Pierre Bezier, Renault, UNISUF system, 1962
A.R. Forrest, Cambridge, 1970
The curves that are now known as Bezier curves were independently developed by P. de Casteljau about 1959 and by P. Bezier about The underlying mathematical theory is based on the concept of Berstein polynomials.
De Casteljau directly exploited this relationshiop; but it was not before 1970 that R. Forrest discovered the connection between Bezier's work and Bernstein polynomials.
Bezier and de Casteljau developed theri theories as part of CAD systems that were being built up at two Frech car companies, Renault and Citroen. The Renault system UNISURF(by Bezier) was soon described in several publications; this is the reason that the underlying theory now bears Bezier's name. Bezier curves and surfaces are now established as the mathematical basis of many CAD systems, they have also become a major tool for the development of new methods for curve and surface descriptions.
Pierre Etienne Bezier was born on September 1, 1910 in Paris. Son and grandson of engineers, he chose this profession too and enrolled to study mechanical engineering at the Ecole des Arts et Metiers and received his degree in In the same year he entered the Ecole Superieure d'Electricite and earnt a second degree in electrical engineering in 1931.
In 1977, 46 years later, he received his DSc degree in mathematics from the University of Paris.
In 1933, aged 23, Bezier entered Renault and worked for this company for 42 years. He started as Tool Setter, became Tool Designer in 1934 and Head of
the Tool Design Office in In 1948, as Director of Production Engineering he was responsible for the design of the transfer lines producing most of the 4 CV mechanical parts. In 1957, he became Director of
Machine Tool Division and was responsible for the automatic assembly of mechanical components, and for the design and production of an NC drilling
and milling machine, most probably one of the first machines in Europe.
Bezier become managing staff member for technical development in 1960 and held this position until 1975 when he retired.
Bezier started his research in CADCAM in 1960 when he devoted a substantial amount of his time working on his UNISURF system. From 1960, his research interest focused on drawing machines, computer control, interactive free-form curve and surface design and 3D milling for manufacturing clay models and masters. His system was launched in 1968 and has been in full
use since 1975 supporting about 1500 staff members today.
Bezier's academic career began in 1968 when he became Professor of Production Engineering at the Conservatoire National des Arts et Metiers.
He held this position until He wrote four books, numerous papers and received several distinctions including the "Steven Anson Coons" of the
Association for Computing Machinery and the "Doctor Honoris Causa" of the Technical University Berlin. He is an honorary member of the American
Society of Mechanical Engineers and of the Societe Belge des Mecaniciens, ex-president of the Societe des Ingenieurs et Scientifiques de France,
Societe des Ingenieurs Arts et Metiers, and he was one of the first Advisory Editors of "Computer-Aided Design".
KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ.,

21. Spline curves J. Ferguson , Boeing Co., 1963
C. de Boor, W. Gordon, General Motors, 1963
to interpolate given data
piecewise polynomial curves with certain differentiability constraints
not to design free form curves
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22. B-spline C. de Boor, 1972 W. Gordon, Richard F. Riesenfeld, 1974
Larry L. Schumaker
Tom Lyche
Nira Dyn
Parametric spline curves, i.e. piecewise polynomial curves with certain differentiability constraints, were first introduced into CAGD by J.Ferguson from Boeing Co. in At the same time C. de Boor and W. Gordon studied these curves at General Motors. Spline curves were only used to interpolate to given data, not to design free form curves. B-spline were initiated by de Boor and others, but they were mostly concerned with approximation theory aspects. Gordon and Riesenfeld married the theory of B-splines with that of Bezier curves and showed that B-spline curves are the proper generalization of Bezier curves.
C. de Boor : Univ. of Wisconsin-Madison
Riesenfeld, Cohen : Utah Univ.
Larry L. Schumaker : Vanderbilt Univ.
Nira Dyn : Tel-Aviv Univ.
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23. Piecewise cubic hermite interpolation
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24. Cubic spline interpolation
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25. Cubic spline interpolation
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26. Spline interpolation based on the 1-norm
Cubic Spline Interpolation with
Natural boundary condition
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27. Condition number
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28. Condition number of B-spline basis
Tom Lyche, Karl Scherer, "On the p-norm Condition Number of Multivariate Triangular Berstein Basis",
10 March 2000
K. Scherer, A.Yu.Shadrin, "New upper bound for the B-spline basis condition number II. A proff of de Boor's 2^k-conjecture",
Tom Lyche, Karl Scherer, "On the Sup-norm Condition Number of Multivariate Triangular Berstein Basis", 1996
C. De Boor, The exact condition of the B-spline basis may be hard to determine", 1990
Tom Lyche and Karl Scherer, On the p-norm condition number of the multivariate triangular Bernstein basis, Journal of Computational and Applied Mathematics 119(2000)
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29. Stability
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30. Blossom
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31. Blossom
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32. B-spline problems Degree Elevation Degree Reduction Knot Insertion
Knot Deletion
Gerald Farin, Curves and Surfaces for Computer Aided Geometric Design, 4th ed, Academic Press (1996)
Ronald N. Goldman, Tom Lyche, editors, Knot Insertion and Deletion Algorithms for B-Spline Curves and Surfaces, SIAM (1993)
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33. Bezier Degree Reduction
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34. Bezier Degree Reduction
Least square method
Legendre-Bernstein basis transformations
Rida T. Farouki, Legendre-Bernstein basis transformations, Journal of Computational and Applied Mathematics 119(2000)
Byung-Gook Lee, Yunbeom Park and Jaechil Yoo, Application of Legendre-Bernstein basis transformations to degree elevation and degree reduction, Computer Aided Geometric Design 19(2002)
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35. Bezier Degree Reduction with constrained
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36. Quasi-interpolants
Tom Lyche and Larry L. Schumaker, Local Spline Approximation Methods, 1974
1.P_d f is local in the sense that depends only on values of f in a small neighboorhood of x
2. reproduce polynomials, spline space
3. the same order as the best spline approximation
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37. Reproduce spline space
This is a linear system of \nu-\mu+d linear equations
m_j unknowns
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38. A cubic quasi-interpolant
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39. Quasi-interpolants local property
the same order as the best spline approximation
can be computed directly without solving systems of equations
Lyche, T. and L. L. Schumaker, Local spline approximation methods, Journal of Approximation Theory 15(1975)
Lyche, T.,L. L. Schumaker and S. Stanley, Quasi-interpolants based on trogonometric splines, Journal of Approximation Theory 95(1998)
KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ.,

40. Contents Affine combination Bezier curves Spline curves
B-spline curves
Condition number
L1-norm spline
Quasi-interpolant
Reference
“Spline Methods Draft”
Tom Lyche and Knut Morken
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