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

Affine combination Linear combinations Affine(Barycentric) combinations Convex combinations Barycentric coordinates KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

Affine combination Euclidean coordinate system Coordinate-free system convex, barycentric combination t0<t<t1 KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

Polynomial interpolation KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

Polynomial interpolation Lagrange polynomials KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

Examples of cubic interpolation not convex hull property KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

Bezier not interpolation convex hull property KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

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 KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

Properties of Bezier Affine invariance Convex hull property Endpoint interpolation Symmetry Linear precision Pseudo-local control Variation Diminishing Property KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

Linear splines KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

Quadratic splines KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

Quadratic splines KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

Representation splines KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

B-spline Recurrence Relation Bernstein polynomial KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

B-spline KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

B-spline Smoothness=Degree-Multiplicity cubic case c2 c1 c0 KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

Spline space KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

Univariate spline KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

Cubic splines KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

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 1962. 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 1930. 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 1945. 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 1979. 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., E-mail:lbg@dongseo.ac.kr

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 KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

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 1963. 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. KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

Piecewise cubic hermite interpolation KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

Cubic spline interpolation KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

Cubic spline interpolation KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

Spline interpolation based on the 1-norm Cubic Spline Interpolation with Natural boundary condition KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

Condition number KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

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) 259-273 KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

Stability KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

Blossom KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

Blossom KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

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) KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

Bezier Degree Reduction KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

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) 145-160. 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) 709-718. KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

Bezier Degree Reduction with constrained KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

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 KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

Reproduce spline space This is a linear system of \nu-\mu+d linear equations m_j unknowns KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

A cubic quasi-interpolant KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

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) 294-325. Lyche, T.,L. L. Schumaker and S. Stanley, Quasi-interpolants based on trogonometric splines, Journal of Approximation Theory 95(1998) 280-309. KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

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 http://kowon.dongseo.ac.kr/~lbg/cagd/ KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr