Download presentation

Presentation is loading. Please wait.

Published byGregory Tatum Modified over 2 years 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

Presentation is loading. Please wait....

OK

Adding Up In Chunks.

Adding Up In Chunks.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on electric meter testing school Ppt on right to information act india Ppt on leadership qualities Authenticity of the bible ppt on how to treat Ppt on project financing in india Appt only ph clinics Ppt on bmc remedy it service Make ppt online Ppt on will world end in 2012 Ppt on abo blood grouping procedure