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

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

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

3/72 Degree Reduction

4/72 Degree Reduction

5/72 Degree Reduction

6/72 Degree Reduction

7/72 Degree Reduction

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

9/72 Least Squares Optimization

10/72 Least Squares Optimization

11/72 Least Squares Optimization

12/72 Least Squares Optimization

13/72 Least Squares Optimization

14/72 Least Squares Optimization

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

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

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

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

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

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

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

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

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

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

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

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

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

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

29/72 Constrained Least Squares Optimization

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

31/72 Constrained Least Squares Optimization

32/72 Constrained Least Squares Optimization

33/72 Constrained Least Squares Optimization

34/72 Constrained Least Squares Optimization

35/72 Constrained Least Squares Optimization

36/72 Least Squares Curves

37/72 Least Squares Curves

38/72 Least Squares Curves

39/72 Least Squares Curves

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

41/72 Degree Reduction

42/72 Degree Reduction

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

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

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

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

47/72 Why Rational Curves? Conics

48/72 Why Rational Curves? Conics

49/72 Why Rational Curves? Conics

50/72 Why Rational Curves? Conics

51/72 Derivatives of Rational Curves

52/72 Derivatives of Rational Curves

53/72 Derivatives of Rational Curves

54/72 Derivatives of Rational Curves

55/72 Splines and Continuity C k continuity:

56/72 Splines and Continuity C k continuity:

57/72 Splines and Continuity C k continuity:

58/72 Splines and Continuity C k continuity:

59/72 Splines and Continuity C k continuity:

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

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

62/72 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 )

63/72 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 ) …

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

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

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

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

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

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!!!

70/72 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. 

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

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

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