Derivative bounds of rational B ézier curves and surfaces

1. Derivative bounds of rational Bézier curves and surfaces Hui-xia Xu Wednesday,Nov. 22, 2006

2. Research background • Bound ofderivative direction can help in detecting intersections between two curves or surfaces • Bound of derivative magnitude can enhance the efficiency of various algorithms for curves and surfaces

3. Methods • Recursive Algorithms • Hodograph and Homogeneous Coordinate • Straightforward Computation

4. Related works(1) • Farin, G., 1983. Algorithms for rational Bézier curves. Computer-Aided Design 15(2), 73-77. • Floater, M.S., 1992. Derivatives of rational Bézier curves. Computer Aided Geometric Design 9(3), 161-174. • Selimovic, I., 2005. New bounds on the magnitude of the derivative of rational Bézier curves and surfaces. Computer Aided Geometric Design 22(4), 321-326. • Zhang, R.-J., Ma, W.-Y., 2006. Some improvements on the derivative bounds of rational Bézier curves and surfaces. Computer Aided Geometric Design 23(7), 563-572.

5. Related works(2) • Sederberg, T.W., Wang, X., 1987. Rational hodographs. Computer Aided Geometric Design 4(4), 333-335. • Hermann, T., 1992. On a tolerance problem of parametric curves and surfaces. Computer Aided Geometric Design 9(2), 109-117. • Satio, T., Wang, G.-J., Sederberg, T.W., 1995. Hodographs and normals of rational curves and surfaces. Computer Aided Geometric Design 12(4), 417-430. • Wang, G.-J., Sederberg, T.W., Satio, T., 1997. Partial derivatives of rational Bézier surfaces. Computer Aided Geometric Design 14(4), 377-381.

6. Related works(3) • Hermann, T., 1999. On the derivatives of second and third degree rational Bézier curves. Computer Aided Geometric Design 16(3), 157-163. • Zhang, R.-J., Wang, G.-J., 2004. The proof of Hermann’s conjecture. Applied Mathematics Letters 17(12), 1387-1390. • Wu, Z., Lin, F., Seah, H.S., Chan, K.Y., 2004. Evaluation of difference bounds for computing rational Bézier curves and surfaces. Computer & Graphics 28(4), 551-558. • Huang, Y.-D., Su, H.-M., 2006. The bound on derivatives of rational Bézier curves. Computer Aided Geometric Design 23(9), 698-702.

7. Derivatives of rational Bézier curves M.S., Floater CAGD 9(1992), 161-174

8. About M.S. Floater • Professor of University of Oslo • Research interests: Geometric modelling, numerical analysis, approximation theory

9. Outline • What to do • The key and innovation points • Main results

10. What to do Recursive Algorithm Two formulas about derivative P'(t) Rational Bézier curve P(t) Two bounds on the derivative magnitude Higher derivatives, curvature and torsion

11. The key andinnovation points

12. Definition • The rational Béziercurve P of degree n as where

13. Recursive algorithm • Defining the intermediate weights and the intermediate points respectively as

14. Recursive algorithm • Computing using the de Casteljau algorithm The former two identities represent the recursive algorithm!

15. Property

16. Derivative formula(1) • The expression of the derivative formula

17. Derivative formula(1) • Rewrite P(t) as where

18. Derivative formula(1) • Rewrite a’(t) and b’(t) as with the principle “accordance with degree”, then after some computation, finally get the derivative formula (1).

19. Derivative formula(2) • The expression of the derivative formula where or

20. Hodograph property

21. Two identities

22. Derivative formula(2) • Rewrite P(t) as • Method of undetermined coefficient

23. Main results

24. Upper bounds(1) where

25. Upper bounds(2) where

26. Some improvements on the derivative bounds of rational Bézier curves and surfaces Ren-Jiang Zhang and Weiyin Ma CAGD23(2006), 563-572

27. About Weiyin Ma • Associate professor of city university of HongKong • Research interests: Computer Aided Geometric Design, CAD/CAM, Virtual Reality for Product Design, Reverse Engineering, Rapid Prototyping and Manufacturing.

28. Outline • What to do • Main results • Innovative points and techniques

29. What to do Hodograph Derivative bound of rational Bézier curves of degree n=2,3 and n=4,5,6 Degree elevation Extension to surfaces Derivative bound of rational Bézier curves of degree n≥2 Recursive algorithm

30. Definition • A rational Bézier curve of degree n is given by • A rational Bézier surface of degree mxn is given by

31. Main results

32. Main results for curves(1) • For every Bézier curve of degree n=2,3 where

33. Main results for curves(2) • For every Bézier curve of degree n=4,5,6 where

34. Main results for curves(3) • For every Bézier curve of degree n≥2 where

35. Main results for surfaces(1) • For every Bézier surface of degree m=2,3

36. Main results for surfaces(2) • For every Bézier surface of degree m=4,5,6

37. Main results for surfaces(3) • For every Bézier surface of degree m≥2 where

38. Innovative points and techniques

39. Innovative points and techniques1 • Represent P’(t) as where

40. Innovative points and techniques1 • Then P’(t) satisfies where

41. Innovative points and techniques1 • Let and are positive numbers, then • and are the same as above, then

42. Innovative points and techniques1 • Let m>0 and then where

43. Proof method • Applying the corresponding innovative points and techniques • In the simplification process based on the principle :

44. Innovative points and techniques2 • Derivative formula(1) • Recursive algorithm

45. About results for curves (3) • Proof the results for curves n≥2 • Point out the result is always stronger than the inequality

46. Results for curves of degree n=7 • The bound for a rational Bézier curve of degree n=7:

47. The bound on derivatives of rational Bézier curves Huang Youdu and Su Huaming CAGD 23(2006), 698-702

48. About authors • Huang Youdu: Professor of Hefei University of Technology , and computation mathematics and computer graphics are his research interests. • Su Huaming: Professor of Hefei University of Technology, and his research interest is computation mathematics.

49. Outline • What to do • The key and techniques • Main results

50. What to do Property of Bernstein New bounds on the curve Rational Bézier curve P(t) Degree elevation On condition some weights are zero Modifying the results