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This research paper explores enhancements in the bounds of derivatives for rational Bézier curves and surfaces, providing a comprehensive review of improvements made by various contributors in the field. The study examines advancements in derivative magnitudes for different degrees of Bézier curves and surfaces, presenting comparisons and proofs of these enhancements. The findings contribute to the better understanding and optimization of computing rational Bézier curves and surfaces.
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The Derivative bounds of rational BézierCurves and Surfaces Pan Yongjuan 2007-3-21
Purpose .
[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.
The derivative bounds of rational Bezier Curves n次有理Bézier 曲线： [Floater 1992] (1) (2)
For n=2,3, the improvements in [Hermann 1999] n =2, (3) (1) (3) is an improvement of (1)!
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
[Wu Zhongke 2004] Evaluation of difference bounds ……, C&G, 28, 551–558. (5)
[Selimovic 2005] New bounds on the magnitude of the derivative of……, CAGD, 22, 321–326. (6) (7)
[Selimovic 2005] The proof of (6) [Floater 1992] [Floater 1992]
[Selimovic 2005] The proof of (7) [Floater 1992]
[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.
[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’)
Lemma let then [Zhang Renjiang 2006] The proof of (8) and (9) [Sederberg and Wang, 1987] where Where,
[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) !
[Zhang Renjiang 2006] The proof of (10) [Floater 1992] [Floater 1992]
[Huang Youdu 2006] The bound on……, CAGD, 23, 698–702. (11) 易证 (11) is an improvement of (6) !
[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) !
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) !
The derivative bounds of rational Bezier Surfaces m×n次有理Bézier 曲面： [Wang 1997] (15) (16)
[Wu Zhongke 2004] Evaluation of difference bounds ……, C&G, 28, 551–558.
[Selimovic 2005] New bounds on the magnitude of the derivative of……, CAGD, 22, 321–326. (17) [Better then (15)] (18) [Better then (16)] (19)
[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)]
