Dr. Scott Schaefer

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
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
LaGrange Polynomials

• Explicit form for interpolating polynomial!

33
Nevilles Algorithm

• Identical to matrix method but uses a geometric
  construction

34
Nevilles Algorithm

• Identical to matrix method but uses a geometric
  construction

35
Nevilles Algorithm

• Identical to matrix method but uses a geometric
  construction

36
Nevilles Algorithm

• Identical to matrix method but uses a geometric
  construction

37
Nevilles Algorithm

• Identical to matrix method but uses a geometric
  construction

38
Nevilles Algorithm

• Identical to matrix method but uses a geometric
  construction

39
Nevilles Algorithm

• Identical to matrix method but uses a geometric
  construction

40
Nevilles Algorithm

• Identical to matrix method but uses a geometric
  construction

41
Nevilles Algorithm

• Identical to matrix method but uses a geometric
  construction

42
Nevilles Algorithm

• Identical to matrix method but uses a geometric
  construction

43
Nevilles Algorithm

• Identical to matrix method but uses a geometric
  construction

44
Nevilles Algorithm

• Identical to matrix method but uses a geometric
  construction

45
Nevilles Algorithm

• Identical to matrix method but uses a geometric
  construction

46
Nevilles Algorithm
47
Nevilles Algorithm
48
Nevilles Algorithm
49
Nevilles Algorithm

• Claim The polynomial produced by Nevilles
  algorithm is unique

50
Nevilles Algorithm

• Claim The polynomial produced by Nevilles
  algorithm is unique
• Proof Assume that there are two degree n
  polynomials such that
• a(ti)b(ti)yi for i0n.

51
Nevilles Algorithm

• Claim The polynomial produced by Nevilles
  algorithm is unique
• Proof Assume that there are two degree n
  polynomials such that
• a(ti)b(ti)yi for i0n.
• c(t)a(t)-b(t) is also a polynomial of degree n

52
Nevilles Algorithm

• Claim The polynomial produced by Nevilles
  algorithm is unique
• Proof Assume that there are two degree n
  polynomials such that
• a(ti)b(ti)yi for i0n.
• c(t)a(t)-b(t) is also a polynomial of degree n
• c(t) has n1 roots at each of the ti

53
Nevilles Algorithm

• Claim The polynomial produced by Nevilles
  algorithm is unique
• Proof Assume that there are two degree n
  polynomials such that
• a(ti)b(ti)yi for i0n.
• c(t)a(t)-b(t) is also a polynomial of degree n
• c(t) has n1 roots at each of the ti
• Polynomials of degree n can have at most n roots!

54
Hermite Interpolation

• Find a polynomial y(t) that interpolates yi,
  yi(1), yi(2), ,
• Always a unique y(t) of degree

55
Hermite Interpolation

• Find a polynomial y(t) that interpolates yi,
  yi(1), yi(2), ,
• Always a unique y(t) of degree

56
Hermite Interpolation

• Find a polynomial y(t) that interpolates yi,
  yi(1), yi(2), ,
• Always a unique y(t) of degree

57
Hermite Interpolation

• Find a polynomial y(t) that interpolates yi,
  yi(1), yi(2), ,
• Always a unique y(t) of degree

58
Hermite Interpolation

• Find a polynomial y(t) that interpolates yi,
  yi(1), yi(2), ,
• Always a unique y(t) of degree

59
Hermite Interpolation

• Find a polynomial y(t) that interpolates yi,
  yi(1), yi(2), ,
• Always a unique y(t) of degree

60
Hermite Interpolation

• Find a polynomial y(t) that interpolates yi,
  yi(1), yi(2), ,
• Always a unique y(t) of degree

61
Hermite Interpolation

• Find a polynomial y(t) that interpolates yi,
  yi(1), yi(2), ,
• Always a unique y(t) of degree