1. 1 Dr. Scott Schaefer Curves and Interpolation

2. 2/61 Smooth Curves How do we create smooth curves?

3. 3/61 Smooth Curves How do we create smooth curves? Parametric curves with polynomials

4. 4/61 Smooth Curves Controlling the shape of the curve

5. 5/61 Smooth Curves Controlling the shape of the curve

6. 6/61 Smooth Curves Controlling the shape of the curve

7. 7/61 Smooth Curves Controlling the shape of the curve

8. 8/61 Smooth Curves Controlling the shape of the curve

9. 9/61 Smooth Curves Controlling the shape of the curve

10. 10/61 Smooth Curves Controlling the shape of the curve

11. 11/61 Smooth Curves Controlling the shape of the curve

12. 12/61 Smooth Curves Controlling the shape of the curve

13. 13/61 Smooth Curves Controlling the shape of the curve Power-basis coefficients not intuitive for controlling shape of curve!!!

14. 14/61 Interpolation Find a polynomial y(t) such that y(t i )=y i

15. 15/61 Interpolation Find a polynomial y(t) such that y(t i )=y i

16. 16/61 Interpolation Find a polynomial y(t) such that y(t i )=y i basis

17. 17/61 Interpolation Find a polynomial y(t) such that y(t i )=y i coefficients

18. 18/61 Interpolation Find a polynomial y(t) such that y(t i )=y i

19. 19/61 Interpolation Find a polynomial y(t) such that y(t i )=y i Vandermonde matrix

20. 20/61 Interpolation Find a polynomial y(t) such that y(t i )=y i

21. 21/61 Interpolation Find a polynomial y(t) such that y(t i )=y i

22. 22/61 Interpolation Find a polynomial y(t) such that y(t i )=y i Intuitive control of curve using “control points”!!!

23. 23/61 Interpolation Perform interpolation for each component separately Combine result to obtain parametric curve

24. 24/61 Interpolation Perform interpolation for each component separately Combine result to obtain parametric curve

25. 25/61 Interpolation Perform interpolation for each component separately Combine result to obtain parametric curve

26. 26/61 Generalized Vandermonde Matrices Assume different basis functions f i (t)

27. 27/61 LaGrange Polynomials Explicit form for interpolating polynomial!

28. 28/61 LaGrange Polynomials Explicit form for interpolating polynomial! 1.522.533.54 -0.2 0.2 0.4 0.6 0.8 1

29. 29/61 1.522.533.54 -0.2 0.2 0.4 0.6 0.8 1 LaGrange Polynomials Explicit form for interpolating polynomial!

30. 30/61 1.522.533.54 -0.2 0.2 0.4 0.6 0.8 1 LaGrange Polynomials Explicit form for interpolating polynomial!

31. 31/61 1.522.533.54 -0.2 0.2 0.4 0.6 0.8 1 LaGrange Polynomials Explicit form for interpolating polynomial!

32. 32/61 LaGrange Polynomials Explicit form for interpolating polynomial!

33. 33/61 Neville’s Algorithm Identical to matrix method but uses a geometric construction

34. 34/61 Neville’s Algorithm Identical to matrix method but uses a geometric construction

35. 35/61 Neville’s Algorithm Identical to matrix method but uses a geometric construction

36. 36/61 Neville’s Algorithm Identical to matrix method but uses a geometric construction

37. 37/61 Neville’s Algorithm Identical to matrix method but uses a geometric construction

38. 38/61 Neville’s Algorithm Identical to matrix method but uses a geometric construction

39. 39/61 Neville’s Algorithm Identical to matrix method but uses a geometric construction

40. 40/61 Neville’s Algorithm Identical to matrix method but uses a geometric construction

41. 41/61 Neville’s Algorithm Identical to matrix method but uses a geometric construction

42. 42/61 Neville’s Algorithm Identical to matrix method but uses a geometric construction

43. 43/61 Neville’s Algorithm Identical to matrix method but uses a geometric construction

44. 44/61 Neville’s Algorithm Identical to matrix method but uses a geometric construction

45. 45/61 Neville’s Algorithm Identical to matrix method but uses a geometric construction

46. 46/61 Neville’s Algorithm

47. 47/61 Neville’s Algorithm

48. 48/61 Neville’s Algorithm

49. 49/61 Neville’s Algorithm Claim: The polynomial produced by Neville’s algorithm is unique

50. 50/61 Neville’s Algorithm Claim: The polynomial produced by Neville’s algorithm is unique Proof: Assume that there are two degree n polynomials such that a(t i )=b(t i )=y i for i=0…n.

51. 51/61 Neville’s Algorithm Claim: The polynomial produced by Neville’s algorithm is unique Proof: Assume that there are two degree n polynomials such that a(t i )=b(t i )=y i for i=0…n. c(t)=a(t)-b(t) is also a polynomial of degree n

52. 52/61 Neville’s Algorithm Claim: The polynomial produced by Neville’s algorithm is unique Proof: Assume that there are two degree n polynomials such that a(t i )=b(t i )=y i for i=0…n. c(t)=a(t)-b(t) is also a polynomial of degree n c(t) has n+1 roots at each of the t i

53. 53/61 Neville’s Algorithm Claim: The polynomial produced by Neville’s algorithm is unique Proof: Assume that there are two degree n polynomials such that a(t i )=b(t i )=y i for i=0…n. c(t)=a(t)-b(t) is also a polynomial of degree n c(t) has n+1 roots at each of the t i Polynomials of degree n can have at most n roots!

54. 54/61 Hermite Interpolation Find a polynomial y(t) that interpolates y i, y i (1), y i (2), …, Always a unique y(t) of degree

55. 55/61 Hermite Interpolation Find a polynomial y(t) that interpolates y i, y i (1), y i (2), …, Always a unique y(t) of degree

56. 56/61 Hermite Interpolation Find a polynomial y(t) that interpolates y i, y i (1), y i (2), …, Always a unique y(t) of degree

57. 57/61 Hermite Interpolation Find a polynomial y(t) that interpolates y i, y i (1), y i (2), …, Always a unique y(t) of degree

58. 58/61 Hermite Interpolation Find a polynomial y(t) that interpolates y i, y i (1), y i (2), …, Always a unique y(t) of degree

59. 59/61 Hermite Interpolation Find a polynomial y(t) that interpolates y i, y i (1), y i (2), …, Always a unique y(t) of degree

60. 60/61 Hermite Interpolation Find a polynomial y(t) that interpolates y i, y i (1), y i (2), …, Always a unique y(t) of degree

61. 61/61 Hermite Interpolation Find a polynomial y(t) that interpolates y i, y i (1), y i (2), …, Always a unique y(t) of degree