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Graphics Programming

Graphics Programming. 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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Graphics Programming

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  1. Graphics Programming Lee Byung-Gook Dongseo Univ. http://kowon.dongseo.ac.kr/~lbg/ Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

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

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

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

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

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

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

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

  10. Bezier • Paul de Faget de Casteljau, Citroen, 1959 • Pierre Bezier, Renault, UNISUF system, 1962 • A.R. Forrest, Cambridge, 1970 Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

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

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

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

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

  15. 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 Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

  16. B-spline • C. de Boor, 1972 • W. Gordon, Richard F. Riesenfeld, 1974 • Larry L. Schumaker • Tom Lyche • Nira Dyn Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

  17. Linear splines Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

  18. Quadratic splines Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

  19. Quadratic splines Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

  20. Representation splines Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

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

  22. B-spline Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

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

  24. B-spline basis functions Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

  25. Refinement relation for B-spline Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

  26. Repeated integration for B-spline Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

  27. Truncated powers for B-spline Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

  28. Cross-sectional Volumes Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

  29. Cross-sectional Volumes for subcubes Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

  30. Box-spline as Cross-sectional Volumes Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

  31. Bivariate Box spline over triangular grid Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

  32. Spline space Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

  33. Univariate spline Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

  34. Cubic splines Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

  35. Condition number Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

  36. 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 Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

  37. Stability Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

  38. Blossom Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

  39. Blossom Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

  40. 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) Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

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