1. A common representation of generalized Ball basis CAGD&CG Seminar Wanqiang Shen 2008-10-22

2. Paper [1] Hongyi Wu. Unifying representation of Bezier curve and generalizedized Ball curves. Applied Mathematics: A Journal of Chinese Universities (Ser. B),2000, 15(1): 109~121. [2] Wanqiang Shen, Guozhao Wang. Extension of the Ball Basis. Journal of Software, 2005, 16(11): 1992-1999. [3] Wanqiang Shen, Guozhao Wang. A Common Representation of Generalized Ball Basis. The 4 th China- Korea Joint Conference on Geometric and Visual Computing & IJCC Workshop, 2008, pp. 3-6.

3. Outline Review Common representation of basis Basis properties Generalized Ball curves Conclusion

4. Review Bernstein Basis

5. Review Ball Basis

6. Review Wang-Ball Basis(Guojin Wang,1987)

7. Review Said-Ball Basis(H.B. Said,1989)

8. Review SBGB(Hongyi Wu,2000) in [1]

9. Review WSGB(Hongyi Wu,2000) in [1]

10. Review Shen&Wang ’ s basis (2005) in [2]

11. [3] Common representation of basis

16. Definition A sequence of functions is a kind of generalized Ball basis of degree n, When, is the Bernstein basis. Where, When and where. are integers satisfying that and. [3] Common representation of basis

17. Properties Positivity Normalization Symmetry Endpoint character Linear independence Common representation

18. Generalized Ball curves Definition Properties:  Convex hull property;  Endpoint interpolation;  Geometric invariance;  Affine invariance.

19. Generalized Ball curves

23. Conclusion Theoretical research Large number of parameters Application is difficult. Future work Application: to fix some parameters and use low degree basis Theory: if it is sufficient and necessary?