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 M ø rken 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 M ø rken 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 B é zier curve

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

11. The Hermite interpolant

14. 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

19. Approximate the rational tensor-product biquadratic B é zier 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 M ø rken, 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

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

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

38. Approximation order New method

39. Approximation order Theorem 4

40. Interpolating higher order derivatives

42. Circle case

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

44. Circle case

45. Restrict n to be odd and place the parameter values symmetrically around t = 0.

46. Circle case

49. Example

50. Thank you Thank you