1. A control polygon scheme for design of planar PH quintic spline curves Francesca Pelosi Maria Lucia Sampoli Rida T. Farouki Carla Manni Speaker:Ying.Liu

2. Abstract Control polygon Knot sequence Pythagorean-hodograph Cubic B-spline curve Control polygon Knot sequence

3. Contents Preparation Definition Why How Single knots: Multiple knots : Others

4. Preparation B-spline curve: (1) (2) (3)

5. Preparation Let n=3, and

6. Preparation

7. Closed curve: Control points ： Knots: For given ， Let overlap and overlap That’s: k=1…n

8. Preparation

9. Definition Polynomial curve r (t)=(x (t),y (t)),satisfies for some polynomial

10. Why Rational offset curves Exact arc length Well-suited real-time CNC interpolator algorithm

11. How( Single knots) Let r (t)=x (t) +i y (t), w (t)=u (t)+ i v (t),

12. How( single knots) The curve interpolates,……, and, is the end point of the curve., and Let

13. How( single knots) Interpolation condition Then (10) End condition For open end condition For closed end condition

14. How( single knots) Nodal points( ): : Open PH Spline curves: Periodic PH Spline curves:

15. How( single knots) Starting approximation: (16) And: (17) Or: (18)

16. How( Multiple knots)

19. Linear precision property Let are double knots, are collinear. Then the curve lie in is a precision line.

20. Linear precision property

22. How( Multiple knots) Local shape modification: Let is to be moved. and are double knots. Then the modified curve is still a PH spline,and well juncture with others.

23. Local shape modification

25. Others Extension to non-uniform knots Closure

26. Thank you!

27. Open PH spline curves Definition: Control points: Knots points: Nodal points: End derivatives:

28. Open PH spline curves

31. Periodic PH spline curves Definition: Control points: Periodic knot sequence, Nodal points: End condition:

32. Periodic PH spline curves

34. Iteration error

35. 90 distinct control points

36. A “randomized” version

37. Iteration error

38. End conditions For open curve: and That is: (12)

39. End conditions For closed curve: That is ： and That is: (13)