1 Dr. Scott Schaefer Curves and Interpolation

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

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

4/61 Smooth Curves Controlling the shape of the curve

5/61 Smooth Curves Controlling the shape of the curve

6/61 Smooth Curves Controlling the shape of the curve

7/61 Smooth Curves Controlling the shape of the curve

8/61 Smooth Curves Controlling the shape of the curve

9/61 Smooth Curves Controlling the shape of the curve

10/61 Smooth Curves Controlling the shape of the curve

11/61 Smooth Curves Controlling the shape of the curve

12/61 Smooth Curves Controlling the shape of the curve

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

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

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

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

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

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

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

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

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

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

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

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

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

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

27/61 LaGrange Polynomials Explicit form for interpolating polynomial!

28/61 LaGrange Polynomials Explicit form for interpolating polynomial!

29/ LaGrange Polynomials Explicit form for interpolating polynomial!

30/ LaGrange Polynomials Explicit form for interpolating polynomial!

31/ LaGrange Polynomials Explicit form for interpolating polynomial!

32/61 LaGrange Polynomials Explicit form for interpolating polynomial!

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

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

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

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

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

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

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

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

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

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

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

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

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

46/61 Neville’s Algorithm

47/61 Neville’s Algorithm

48/61 Neville’s Algorithm

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

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.

51/61 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

52/61 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

53/61 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!

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

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

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

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

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

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

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