Rational curves interpolated by polynomial curves

1. Rational curves interpolated by polynomial curves Reporter Lian Zhou Sep. 21 2006

2. Introduction • De Boor et al.,1987 • Dokken et al.,1990 • Floater,1997 • Goladapp,1991 • Garndine and Hogan,2004

3. Introduction • Jaklic et al.,Preprint • Lyche and Mørken 1994 • Morken and Scherer 1997 • Schaback 1998 • Floater 2006

4. Introduction • Non-vanishing curvature of the curve De Boor et al.,1987 • Circle Dekken et al.,1990;Goldapp,1991; Lyche and Mørken 1994

5. Introduction • Conic section Fang, 1999;Floater, 1997

6. Introduction • de Boor et al., 1987 where a 6th-order accurate cubic interpolation scheme for planar curves was constructed.

7. Introduction • Lian Fang 1999

8. Introduction

9. The Hermite interpolant • We will approximate the rational quadratic Bézier curve

10. The Hermite interpolant • ellipse when w < l • parabola when w = 1 • hyperbola when w >1;

11. The Hermite interpolant

12. The Hermite interpolant

13. The Hermite interpolant

14. The Hermite interpolant • Lemma 1

15. The Hermite interpolant • Lemma 2

16. The Hermite interpolant • Theorem 1 The curve q has a total number of 2n contacts with r since the equation f(q(t)) = 0 has 2n roots inside [0, 1].

17. The Hermite interpolant

18. The Hermite interpolant

19. The Hermite interpolant • Approximate the rational tensor-product biquadratic Bézier surface

20. Disadvantage • For general m, little seems to be known about the existence of such interpolants apart from the two families of interpolants of odd degree m to circles and conic sections found in (Lyche and Mørken, 1994) and (Floater, 1997), each having a total of 2m contacts.

21. High order approximation of rational curves by polynomial curves Michael S. Floater Computer Aided Geometric Design 23 (2006) 621–628

22. Method • Let be the rational curve r(t)=f(t)/g(t).

23. Two assumptions

24. Basic idea

25. Basic idea

26. Basic idea

27. Basic idea

28. Basic idea

29. Basic idea

30. Theorem 3 • There are unique polynomials X and Y of degrees at most N − 1 and n + N − 2, respectively, that solve (4). With these X and Y , p in (5) is a polynomial of degree at most n+k −2 that solves (1).

31. Euclid’s g.c.d.gorithm • Now describe how Euclid’s algorithm can be used to find the solutions X and Y .

32. Euclid’s g.c.d.gorithm

33. Euclid’s g.c.d.gorithm

34. Euclid’s g.c.d.gorithm

35. Euclid’s g.c.d.gorithm

36. Euclid’s g.c.d.gorithm

37. Approximation order • Algebraic form of circle or conic section Dokken et al., 1990; Goldapp, 1991; Lyche and Mørken, 1994; Floater, 1997;

38. Approximation order • New method

39. Approximation order • Theorem 4

40. Interpolating higher order derivatives

41. Interpolating higher order derivatives

42. Circle case

43. Circle case • Add the vector (1, 0) to (15) Then

44. Circle case

45. Circle case • Restrict nto be odd and place the parameter values symmetrically around t= 0.

46. Circle case

47. Circle case

48. Circle case

49. Example

50. Thank you