Curves and Interpolation

1. Curves and Interpolation Dr. Scott Schaefer

2. Smooth Curves • How do we create smooth curves?

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

4. Smooth Curves • Controlling the shape of the curve

5. Smooth Curves • Controlling the shape of the curve

6. Smooth Curves • Controlling the shape of the curve

7. Smooth Curves • Controlling the shape of the curve

8. Smooth Curves • Controlling the shape of the curve

9. Smooth Curves • Controlling the shape of the curve

10. Smooth Curves • Controlling the shape of the curve

11. Smooth Curves • Controlling the shape of the curve

12. Smooth Curves • Controlling the shape of the curve

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

14. Interpolation • Find a polynomial y(t) such that y(ti)=yi

15. Interpolation • Find a polynomial y(t) such that y(ti)=yi

16. Interpolation • Find a polynomial y(t) such that y(ti)=yi basis

17. Interpolation • Find a polynomial y(t) such that y(ti)=yi coefficients

18. Interpolation • Find a polynomial y(t) such that y(ti)=yi

19. Interpolation • Find a polynomial y(t) such that y(ti)=yi Vandermonde matrix

20. Interpolation • Find a polynomial y(t) such that y(ti)=yi

21. Interpolation • Find a polynomial y(t) such that y(ti)=yi

22. Interpolation • Find a polynomial y(t) such that y(ti)=yi Intuitive control of curve using “control points”!!!

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

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

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

26. Generalized Vandermonde Matrices • Assume different basis functions fi(t)

27. LaGrange Polynomials • Explicit form for interpolating polynomial!

28. 1 0.8 0.6 0.4 0.2 1.5 2 2.5 3 3.5 4 -0.2 LaGrange Polynomials • Explicit form for interpolating polynomial!

29. 1 0.8 0.6 0.4 0.2 1.5 2 2.5 3 3.5 4 -0.2 LaGrange Polynomials • Explicit form for interpolating polynomial!

30. 1 0.8 0.6 0.4 0.2 1.5 2 2.5 3 3.5 4 -0.2 LaGrange Polynomials • Explicit form for interpolating polynomial!

31. 1 0.8 0.6 0.4 0.2 1.5 2 2.5 3 3.5 4 -0.2 LaGrange Polynomials • Explicit form for interpolating polynomial!

32. LaGrange Polynomials • Explicit form for interpolating polynomial!

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

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

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

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

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

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

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

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

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

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

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

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

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

46. Neville’s Algorithm

47. Neville’s Algorithm

48. Neville’s Algorithm

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

50. 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(ti)=b(ti)=yi for i=0…n.