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Spline Methods in CAGD

Spline Methods in CAGD. Lee Byung-Gook Dongseo Univ. http://kowon.dongseo.ac.kr/~lbg/. Affine combination. Linear combinations Affine(Barycentric) combinations Convex combinations Barycentric coordinates. Affine combination. Euclidean coordinate system. Coordinate-free system.

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Spline Methods in CAGD

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  1. 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

  2. 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

  3. Affine combination Euclidean coordinate system Coordinate-free system KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

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

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

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

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

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

  9. 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

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

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

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

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

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

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

  16. B-spline • Smoothness=Degree-Multiplicity KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

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

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

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

  20. 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

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

  22. 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

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

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

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

  26. 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

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

  28. 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

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

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

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

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

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

  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) 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

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

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

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

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

  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) 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

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

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