Download presentation

Presentation is loading. Please wait.

Published byGregory Tatum Modified about 1 year ago

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

Similar presentations

© 2016 SlidePlayer.com Inc.

All rights reserved.

Ads by Google