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1 Dr. Scott Schaefer Least Squares Curves, Rational Representations, Splines and Continuity.

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Presentation on theme: "1 Dr. Scott Schaefer Least Squares Curves, Rational Representations, Splines and Continuity."— Presentation transcript:

1 1 Dr. Scott Schaefer Least Squares Curves, Rational Representations, Splines and Continuity

2 2/72 Degree Reduction Given a set of coefficients for a Bezier curve of degree n+1, find the best set of coefficients of a Bezier curve of degree n that approximate that curve

3 3/72 Degree Reduction

4 4/72 Degree Reduction

5 5/72 Degree Reduction

6 6/72 Degree Reduction

7 7/72 Degree Reduction

8 8/72 Degree Reduction Problem: end-points are not interpolated

9 9/72 Least Squares Optimization

10 10/72 Least Squares Optimization

11 11/72 Least Squares Optimization

12 12/72 Least Squares Optimization

13 13/72 Least Squares Optimization

14 14/72 Least Squares Optimization

15 15/72 The PseudoInverse What happens when isn’t invertible?

16 16/72 The PseudoInverse What happens when isn’t invertible?

17 17/72 The PseudoInverse What happens when isn’t invertible?

18 18/72 The PseudoInverse What happens when isn’t invertible?

19 19/72 The PseudoInverse What happens when isn’t invertible?

20 20/72 The PseudoInverse What happens when isn’t invertible?

21 21/72 The PseudoInverse What happens when isn’t invertible?

22 22/72 The PseudoInverse What happens when isn’t invertible?

23 23/72 The PseudoInverse What happens when isn’t invertible?

24 24/72 The PseudoInverse What happens when isn’t invertible?

25 25/72 The PseudoInverse What happens when isn’t invertible?

26 26/72 The PseudoInverse What happens when isn’t invertible?

27 27/72 The PseudoInverse What happens when isn’t invertible?

28 28/72 The PseudoInverse What happens when isn’t invertible?

29 29/72 Constrained Least Squares Optimization

30 30/72 Constrained Least Squares Optimization Constraint Space Error Function F(x) Solution

31 31/72 Constrained Least Squares Optimization

32 32/72 Constrained Least Squares Optimization

33 33/72 Constrained Least Squares Optimization

34 34/72 Constrained Least Squares Optimization

35 35/72 Constrained Least Squares Optimization

36 36/72 Least Squares Curves

37 37/72 Least Squares Curves

38 38/72 Least Squares Curves

39 39/72 Least Squares Curves

40 40/72 Degree Reduction Problem: end-points are not interpolated

41 41/72 Degree Reduction

42 42/72 Degree Reduction

43 43/72 Rational Curves Curves defined in a higher dimensional space that are “projected” down

44 44/72 Rational Curves Curves defined in a higher dimensional space that are “projected” down

45 45/72 Rational Curves Curves defined in a higher dimensional space that are “projected” down

46 46/72 Rational Curves Curves defined in a higher dimensional space that are “projected” down

47 47/72 Why Rational Curves? Conics

48 48/72 Why Rational Curves? Conics

49 49/72 Why Rational Curves? Conics

50 50/72 Why Rational Curves? Conics

51 51/72 Derivatives of Rational Curves

52 52/72 Derivatives of Rational Curves

53 53/72 Derivatives of Rational Curves

54 54/72 Derivatives of Rational Curves

55 55/72 Splines and Continuity C k continuity:

56 56/72 Splines and Continuity C k continuity:

57 57/72 Splines and Continuity C k continuity:

58 58/72 Splines and Continuity C k continuity:

59 59/72 Splines and Continuity C k continuity:

60 60/72 Splines and Continuity Assume two Bezier curves with control points p 0,…,p n and q 0,…,q m

61 61/72 Splines and Continuity Assume two Bezier curves with control points p 0,…,p n and q 0,…,q m C 0 : p n =q 0

62 62/72 Splines and Continuity Assume two Bezier curves with control points p 0,…,p n and q 0,…,q m C 0 : p n =q 0 C 1 : n(p n -p n-1 )=m(q 1 -q 0 )

63 63/72 Splines and Continuity Assume two Bezier curves with control points p 0,…,p n and q 0,…,q m C 0 : p n =q 0 C 1 : n(p n -p n-1 )=m(q 1 -q 0 ) C 2 : n(n-1)(p n -2p n-1 +p n-2 )=m(m-1)(q 0 -2q 1 +q 2 ) …

64 64/72 Splines and Continuity Geometric Continuity  A curve is G k if there exists a reparametrization such that the curve is C k

65 65/72 Splines and Continuity Geometric Continuity  A curve is G k if there exists a reparametrization such that the curve is C k

66 66/72 Splines and Continuity Geometric Continuity  A curve is G k if there exists a reparametrization such that the curve is C k

67 67/72 Problems with Bezier Curves More control points means higher degree Moving one control point affects the entire curve

68 68/72 Problems with Bezier Curves More control points means higher degree Moving one control point affects the entire curve

69 69/72 Problems with Bezier Curves More control points means higher degree Moving one control point affects the entire curve Solution: Use lots of Bezier curves and maintain C k continuity!!!

70 70/72 Problems with Bezier Curves More control points means higher degree Moving one control point affects the entire curve Solution: Use lots of Bezier curves and maintain C k continuity!!! Difficult to keep track of all the constraints. 

71 71/72 B-spline Curves Not a single polynomial, but lots of polynomials that meet together smoothly Local control

72 72/72 B-spline Curves Not a single polynomial, but lots of polynomials that meet together smoothly Local control


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