Spline Methods in CAGD Lee ByungGook Dongseo Univ

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

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

Representation Bezier 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

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 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 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 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

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 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

B-spline problems • • Degree Elevation Degree Reduction Knot Insertion Knot Deletion Gerald Farin, Curves and Surfaces for Computer Aided Geometric Design, 4 th ed, Academic Press (1996) Ronald N. Goldman, Tom Lyche, editors, Knot Insertion and Deletion Algorithms for BSpline 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 KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ. , E-mail: lbg@dongseo. ac. kr

Reproduce spline space 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 L 1 -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