The Derivative bounds of rational B é zier Curves and Surfaces

1. The Derivative bounds of rational BézierCurves and Surfaces Pan Yongjuan 2007-3-21

2. Purpose .

3. [Sederberg 1987] Rational hodographs. CAGD,4 (4), 333–335. • [Floater 1992] Derivatives of rational Bézier curves. CAGD, 9, 161–174. • [Wang 1997]Partial derivatives of rational Bézier surfaces. CAGD 14 (4), 377–381. • [Hermann 1999] On the derivatives of second and third degree rational Bézier curve. CAGD,16, 157–163. • [Wu Zhongke 2004] Evaluation of difference bounds for computing rational Bézier curves and surfaces. C&G, 28, 551–558. • [Selimovic 2005] New bounds on the magnitude of the derivative of rational Bézier curves and surfaces. CAGD, 22, 321–326. • [Zhang Renjiang 2006] Some improvements on the derivative bounds of rational Bézier curves and surfaces. CAGD, 23, 563–572. • [Huang Youdu 2006] The bound on derivatives of rational Bézier curves. CAGD, 23, 698–702.

4. The derivative bounds of rational Bezier Curves n次有理Bézier 曲线： [Floater 1992] (1) (2)

5. For n=2,3, the improvements in [Hermann 1999] n =2, (3) (1) (3) is an improvement of (1)!

6. For n=2,3, the improvements in [Hermann 1999] n =3, (4) (4) is an improvement of (2)! [Zhang Renjiang 2004] Applied Mathematics Letters, 17, 1387-1390

7. [Wu Zhongke 2004] Evaluation of difference bounds ……, C&G, 28, 551–558. (5)

8. [Selimovic 2005] New bounds on the magnitude of the derivative of……, CAGD, 22, 321–326. (6) (7)

9. [Selimovic 2005] The proof of (6) [Floater 1992] [Floater 1992]

10. [Selimovic 2005] The proof of (7) [Floater 1992]

11. [Selimovic 2005] Comparison to [Floater1992] (1) [Floater 1992] (2) (6) (7) (6) is an improvement of (1)! Neither (7) nor (2) is stronger than the other.

12. [Zhang Renjiang 2006] Some improvements on……, CAGD, 23, 563–572. When n =2, 3 (8) (8) is an improvement of (6) and (7)! When n = 4, 5, 6 (9) (9) is an improvement of (7) and (2)! (9’)

13. Lemma let then [Zhang Renjiang 2006] The proof of (8) and (9) [Sederberg and Wang, 1987] where Where,

14. [Zhang Renjiang 2006] Some improvements on……, CAGD, 23, 563–572. When n = 7 (9”) For all n, (10) where (10) is an improvement of (6) !

15. [Zhang Renjiang 2006] The proof of (10) [Floater 1992] [Floater 1992]

16. [Huang Youdu 2006] The bound on……, CAGD, 23, 698–702. (11) 易证 (11) is an improvement of (6) !

17. [Huang Youdu 2006] The bound on……, CAGD, 23, 698–702. Let then (11) Modifying the result with degree elevation, Let (12) (12) is an improvement of (11) !

18. Dealing with the case when some are zero [Huang Youdu 2006] The bound on……, CAGD, 23, 698–702. (13) (14) (14) is an improvement of (13) !

19. The derivative bounds of rational Bezier Surfaces m×n次有理Bézier 曲面： [Wang 1997] (15) (16)

20. [Wu Zhongke 2004] Evaluation of difference bounds ……, C&G, 28, 551–558.

21. [Selimovic 2005] New bounds on the magnitude of the derivative of……, CAGD, 22, 321–326. (17) [Better then (15)] (18) [Better then (16)] (19)

22. [Zhang Renjiang 2006] Some improvements on……, CAGD, 23, 563–572. m=2, 3 (20) [Better then (17)] m=4, 5,6 (21) [Better then (19)] (22) [Better then (18)]

23. Thank you!