1. the Forum of IMS in Ewha Univ. May 27. 2003, Byung-Gook Lee, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Mathematics in Computer Graphics Byung-Gook Lee Dongseo Univ. http://kowon.dongseo.ac.kr/~lbg/

2. the Forum of IMS in Ewha Univ. May 27. 2003, Byung-Gook Lee, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

3. Computer Graphics The generation of graphical output using a computer by developing software to accomplish the task by using pre-existing application software like Photoshop, 3D Studio Max, Maya, …

4. the Forum of IMS in Ewha Univ. May 27. 2003, Byung-Gook Lee, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Computer Graphics Modeling Animation Rendering

5. the Forum of IMS in Ewha Univ. May 27. 2003, Byung-Gook Lee, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Modeling The creation of mathematical models of 2D and 3D objects in the 3D environment of a computer.

6. the Forum of IMS in Ewha Univ. May 27. 2003, Byung-Gook Lee, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Animation Topics include traditional principles of animation, kinematic and dynamic modeling techniques, physical simulation, procedural methods, and motion capture based animation.

7. the Forum of IMS in Ewha Univ. May 27. 2003, Byung-Gook Lee, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Rendering These models, starting as a wire frame model, are digitally wrapped with textures and rendered with reflections, transparencies, and shadows to give a photo-realistic view of the object or building.

8. the Forum of IMS in Ewha Univ. May 27. 2003, Byung-Gook Lee, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Web 3D

9. the Forum of IMS in Ewha Univ. May 27. 2003, Byung-Gook Lee, Dongseo Univ., E-mail:lbg@dongseo.ac.kr CAGD Computer Aided Geometric Design  interpolation  spline  bezier  B-spline  NURBS  subdivision  simplification  parametrization CAGD is a branch of applied mathematics concerned with algorithms for the design of smooth curves and surfaces and for their efficients mathematical representation.

10. the Forum of IMS in Ewha Univ. May 27. 2003, Byung-Gook Lee, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Affine combination Linear combinations Affine(Barycentric) combinations Convex combinations Barycentric coordinates

11. the Forum of IMS in Ewha Univ. May 27. 2003, Byung-Gook Lee, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Affine combination Euclidean coordinate system Coordinate-free system

12. the Forum of IMS in Ewha Univ. May 27. 2003, Byung-Gook Lee, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Polynomial interpolation

13. the Forum of IMS in Ewha Univ. May 27. 2003, Byung-Gook Lee, Dongseo Univ., E-mail:lbg@dongseo.ac.kr General polynomial Interpolation Lagrange polynomials

14. the Forum of IMS in Ewha Univ. May 27. 2003, Byung-Gook Lee, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Examples of cubic interpolation

15. the Forum of IMS in Ewha Univ. May 27. 2003, Byung-Gook Lee, 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

16. the Forum of IMS in Ewha Univ. May 27. 2003, Byung-Gook Lee, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Bezier

17. the Forum of IMS in Ewha Univ. May 27. 2003, Byung-Gook Lee, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Representation Bezier

18. the Forum of IMS in Ewha Univ. May 27. 2003, Byung-Gook Lee, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Properties of Bezier Affine invariance Convex hull property Endpoint interpolation Symmetry Linear precision Pseudo-local control

19. the Forum of IMS in Ewha Univ. May 27. 2003, Byung-Gook Lee, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Pierre BEZIER died on November 25, 1999. The following short biography has been published in volume 22, number 9 (november 1990) of "Computer Aided Design", a special issue devoted to Bezier techniques. 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 responsable 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 responsable 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 manufactoring 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". Christophe Rabut, Departement de Genie Mathematique Institut National des Sciences Appliquees 135, avenue de Rangueil F-31077 TOULOUSE Cedex 4 (France)

20. the Forum of IMS in Ewha Univ. May 27. 2003, Byung-Gook Lee, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Bezier Degree Reduction

21. the Forum of IMS in Ewha Univ. May 27. 2003, Byung-Gook Lee, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Spline curve 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

22. the Forum of IMS in Ewha Univ. May 27. 2003, Byung-Gook Lee, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Piecewise cubic hermite interpolation

23. the Forum of IMS in Ewha Univ. May 27. 2003, Byung-Gook Lee, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Cubic spline interpolation

24. the Forum of IMS in Ewha Univ. May 27. 2003, Byung-Gook Lee, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Natural boundary condition

25. the Forum of IMS in Ewha Univ. May 27. 2003, Byung-Gook Lee, 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 Cohen

26. the Forum of IMS in Ewha Univ. May 27. 2003, Byung-Gook Lee, Dongseo Univ., E-mail:lbg@dongseo.ac.kr B-spline

27. the Forum of IMS in Ewha Univ. May 27. 2003, Byung-Gook Lee, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Linear splines

28. the Forum of IMS in Ewha Univ. May 27. 2003, Byung-Gook Lee, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Quadratic splines

29. the Forum of IMS in Ewha Univ. May 27. 2003, Byung-Gook Lee, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Quadratic splines

30. the Forum of IMS in Ewha Univ. May 27. 2003, Byung-Gook Lee, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Representation splines

31. the Forum of IMS in Ewha Univ. May 27. 2003, Byung-Gook Lee, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Cubic splines

32. the Forum of IMS in Ewha Univ. May 27. 2003, Byung-Gook Lee, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Spline problems Degree Elevation Degree Reduction Knot Insertion Knot Deletion

33. the Forum of IMS in Ewha Univ. May 27. 2003, Byung-Gook Lee, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Subdivision schemes

34. the Forum of IMS in Ewha Univ. May 27. 2003, Byung-Gook Lee, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Subdivision Surfaces

35. the Forum of IMS in Ewha Univ. May 27. 2003, Byung-Gook Lee, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Subdivision Surfaces Tony Derose Pixar Animation Studios Geri’s game

36. the Forum of IMS in Ewha Univ. May 27. 2003, Byung-Gook Lee, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Quasi-interpolants

37. the Forum of IMS in Ewha Univ. May 27. 2003, Byung-Gook Lee, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Polygonal Simplification

38. the Forum of IMS in Ewha Univ. May 27. 2003, Byung-Gook Lee, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Multiresolution

39. the Forum of IMS in Ewha Univ. May 27. 2003, Byung-Gook Lee, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Parametrization 3D mesh parameterization with fixed boundary

40. the Forum of IMS in Ewha Univ. May 27. 2003, Byung-Gook Lee, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Texture Mapping

41. the Forum of IMS in Ewha Univ. May 27. 2003, Byung-Gook Lee, Dongseo Univ., E-mail:lbg@dongseo.ac.kr Image Compression

42. the Forum of IMS in Ewha Univ. May 27. 2003, Byung-Gook Lee, Dongseo Univ., E-mail:lbg@dongseo.ac.kr References Curves and Surfaces for Computer Aided Geometric Design, Gerald Farin, Academic Press The NURBS book, Les Piegl and Wayne Tiller, Springer Spline Methods Drafts, Tom Lyche and Knut Morken Computer Graphics & Geometric Modeling, Davis Salomon, Springer http://kowon.dongseo.ac.kr/~lbg/cagd/