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