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

2. 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. Degree Reduction

4. Degree Reduction

5. Degree Reduction

6. Degree Reduction

7. Degree Reduction

8. Degree Reduction
Problem: end-points are not interpolated

9. Least Squares Optimization

10. Least Squares Optimization

11. Least Squares Optimization

12. Least Squares Optimization

13. Least Squares Optimization

14. Least Squares Optimization

15. The PseudoInverse
What happens when isn’t invertible?

16. The PseudoInverse
What happens when isn’t invertible?

17. The PseudoInverse
What happens when isn’t invertible?

18. The PseudoInverse
What happens when isn’t invertible?

19. The PseudoInverse
What happens when isn’t invertible?

20. The PseudoInverse
What happens when isn’t invertible?

21. The PseudoInverse
What happens when isn’t invertible?

22. The PseudoInverse
What happens when isn’t invertible?

23. The PseudoInverse
What happens when isn’t invertible?

24. The PseudoInverse
What happens when isn’t invertible?

25. The PseudoInverse
What happens when isn’t invertible?

26. The PseudoInverse
What happens when isn’t invertible?

27. The PseudoInverse
What happens when isn’t invertible?

28. The PseudoInverse
What happens when isn’t invertible?

29. Constrained Least Squares Optimization

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

31. Constrained Least Squares Optimization

32. Constrained Least Squares Optimization

33. Constrained Least Squares Optimization

34. Constrained Least Squares Optimization

35. Constrained Least Squares Optimization

36. Least Squares Curves

37. Least Squares Curves

38. Least Squares Curves

39. Least Squares Curves

40. Degree Reduction
Problem: end-points are not interpolated

41. Degree Reduction

42. Degree Reduction

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

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

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

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

47. Why Rational Curves?
Conics

48. Why Rational Curves?
Conics

49. Why Rational Curves?
Conics

50. Why Rational Curves?
Conics

51. Derivatives of Rational Curves

52. Derivatives of Rational Curves

53. Derivatives of Rational Curves

54. Derivatives of Rational Curves

55. Splines and Continuity
Ck continuity:

56. Splines and Continuity
Ck continuity:

57. Splines and Continuity
Ck continuity:

58. Splines and Continuity
Ck continuity:

59. Splines and Continuity
Ck continuity:

60. Splines and Continuity
Assume two Bezier curves with control points p0,…,pn and q0,…,qm

61. Splines and Continuity
Assume two Bezier curves with control points p0,…,pn and q0,…,qm
C0: pn=q0

62. Splines and Continuity
Assume two Bezier curves with control points p0,…,pn and q0,…,qm
C0: pn=q0
C1: n(pn-pn-1)=m(q1-q0)

63. Splines and Continuity
Assume two Bezier curves with control points p0,…,pn and q0,…,qm
C0: pn=q0
C1: n(pn-pn-1)=m(q1-q0)
C2: n(n-1)(pn-2pn-1+pn-2)=m(m-1)(q0-2q1+q2) …

64. Splines and Continuity
Geometric Continuity
A curve is Gk if there exists a reparametrization such that the curve is Ck

65. Splines and Continuity
Geometric Continuity
A curve is Gk if there exists a reparametrization such that the curve is Ck

66. Splines and Continuity
Geometric Continuity
A curve is Gk if there exists a reparametrization such that the curve is Ck

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

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

69. 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 Ck continuity!!!

70. 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 Ck continuity!!!
Difficult to keep track of all the constraints. 

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

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