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Dr. Scott Schaefer

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Lagrange Interpolation Bezier Curve Bezier Curve Bezier Curve Bezier Curve Bezier Curve Bezier Curve Lagrange Interpolation Another Bezier curve of vectors!!! – PowerPoint PPT presentation

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Title: Dr. Scott Schaefer


1
The Bernstein Basis and Bezier Curves
  • Dr. Scott Schaefer

2
Problems with Interpolation
3
Problems with Interpolation
4
Bezier Curves
  • Polynomial curves that seek to approximate rather
    than to interpolate

5
Bernstein Polynomials
  • Degree 1 (1-t), t
  • Degree 2 (1-t)2, 2(1-t)t, t2
  • Degree 3 (1-t)3, 3(1-t)2t, 3(1-t)t2, t3

6
Bernstein Polynomials
  • Degree 1 (1-t), t
  • Degree 2 (1-t)2, 2(1-t)t, t2
  • Degree 3 (1-t)3, 3(1-t)2t, 3(1-t)t2, t3
  • Degree 4 (1-t)4, 4(1-t)3t, 6(1-t)2t2,
    4(1-t)t3,t4

7
Bernstein Polynomials
  • Degree 1 (1-t), t
  • Degree 2 (1-t)2, 2(1-t)t, t2
  • Degree 3 (1-t)3, 3(1-t)2t, 3(1-t)t2, t3
  • Degree 4 (1-t)4, 4(1-t)3t, 6(1-t)2t2,
    4(1-t)t3,t4
  • Degree 5 (1-t)5, 5(1-t)4t, 10(1-t)3t2,
    10(1-t)2t3 ,5(1-t)t4,t5
  • Degree n for

8
Properties of Bernstein Polynomials

9
Properties of Bernstein Polynomials

10
Properties of Bernstein Polynomials

11
Properties of Bernstein Polynomials

Binomial Theorem
12
Properties of Bernstein Polynomials

Binomial Theorem
13
Properties of Bernstein Polynomials

Binomial Theorem
14
Properties of Bernstein Polynomials

Binomial Theorem
15
Properties of Bernstein Polynomials

16
Properties of Bernstein Polynomials

17
Properties of Bernstein Polynomials

18
Properties of Bernstein Polynomials

19
More Properties of Bernstein Polynomials

20
More Properties of Bernstein Polynomials

21
More Properties of Bernstein Polynomials

22
More Properties of Bernstein Polynomials

23
Properties of Bernstein Polynomials
24
Properties of Bernstein Polynomials
  • Base case

25
Properties of Bernstein Polynomials
  • Base case

26
Properties of Bernstein Polynomials
  • Base case
  • Inductive Step Assume

27
Properties of Bernstein Polynomials
  • Base case
  • Inductive Step Assume

28
Properties of Bernstein Polynomials
  • Base case
  • Inductive Step Assume

29
Properties of Bernstein Polynomials
  • Base case
  • Inductive Step Assume

30
Bezier Curves
31
Bezier Curves
32
Bezier Curves
33
Bezier Curve Properties
  • Interpolate end-points

34
Bezier Curve Properties
  • Interpolate end-points

35
Bezier Curve Properties
  • Interpolate end-points

36
Bezier Curve Properties
  • Interpolate end-points

37
Bezier Curve Properties
  • Interpolate end-points
  • Tangent at end-points in direction of first/last
    edge

38
Bezier Curve Properties
  • Interpolate end-points
  • Tangent at end-points in direction of first/last
    edge

39
Bezier Curve Properties
  • Interpolate end-points
  • Tangent at end-points in direction of first/last
    edge

40
Bezier Curve Properties
  • Interpolate end-points
  • Tangent at end-points in direction of first/last
    edge

41
Bezier Curve Properties
  • Interpolate end-points
  • Tangent at end-points in direction of first/last
    edge

42
Bezier Curve Properties
  • Interpolate end-points
  • Tangent at end-points in direction of first/last
    edge

Another Bezier curve of vectors!!!
43
Bezier Curve Properties
  • Interpolate end-points
  • Tangent at end-points in direction of first/last
    edge

44
Bezier Curve Properties
  • Interpolate end-points
  • Tangent at end-points in direction of first/last
    edge
  • Curve lies within the convex hull of the control
    points

45
Bezier Curve Properties
  • Interpolate end-points
  • Tangent at end-points in direction of first/last
    edge
  • Curve lies within the convex hull of the control
    points

Bezier
Lagrange
46
Matrix Form of Bezier Curves
47
Matrix Form of Bezier Curves
48
Matrix Form of Bezier Curves
49
Matrix Form of Bezier Curves
50
Matrix Form of Bezier Curves
Computation in monomial basis is unstable!!!
Most proofs/computations are easier in Bernstein
basis!!!
51
Change of Basis
Bezier coefficients
52
Change of Basis
Coefficients in monomial basis!!!
Bezier coefficients
53
Change of Basis
54
Change of Basis
55
Change of Basis
56
Change of Basis
57
Change of Basis
monomial coefficients
58
Change of Basis
Coefficients in Bezier basis!!!
monomial coefficients
59
Degree Elevation
  • Power basis is trivial add 0 tn1
  • What about Bezier basis?

60
Degree Elevation
61
Degree Elevation
62
Degree Elevation
63
Degree Elevation
64
Degree Elevation
65
Degree Elevation
66
Degree Elevation
67
Degree Elevation
68
Degree Elevation
69
Degree Elevation
70
Degree Elevation
71
Pyramid Algorithms for Bezier Curves
  • Polynomials arent pretty
  • Is there an easier way to evaluate the equation
    of a Bezier curve?

72
Pyramid Algorithms forBezier Curves
73
Pyramid Algorithms forBezier Curves
74
Pyramid Algorithms forBezier Curves
75
Pyramid Algorithms forDerivatives of Bezier
Curves
Take derivative of any level of pyramid!!! (up to
constant multiple)
76
Pyramid Algorithms forDerivatives of Bezier
Curves
Take derivative of any level of pyramid!!! (up to
constant multiple)
77
Subdividing Bezier Curves
  • Given a single Bezier curve, construct two
    smaller Bezier curves whose union is exactly the
    original curve

78
Subdividing Bezier Curves
  • Given a single Bezier curve, construct two
    smaller Bezier curves whose union is exactly the
    original curve

79
Subdividing Bezier Curves
  • Given a single Bezier curve, construct two
    smaller Bezier curves whose union is exactly the
    original curve

80
Subdividing Bezier Curves
  • Given a single Bezier curve, construct two
    smaller Bezier curves whose union is exactly the
    original curve

81
Subdividing Bezier Curves
  • Given a single Bezier curve, construct two
    smaller Bezier curves whose union is exactly the
    original curve

82
Subdividing Bezier Curves
Control points for left curve!!!
83
Subdividing Bezier Curves
Control points for right curve!!!
84
Subdividing Bezier Curves
85
Variation Diminishing
  • Intuitively means that the curve wiggles no
    more than its control polygon

86
Variation Diminishing
  • Intuitively means that the curve wiggles no
    more than its control polygon

Lagrange Interpolation
87
Variation Diminishing
  • Intuitively means that the curve wiggles no
    more than its control polygon

Bezier Curve
88
Variation Diminishing
  • For any line, the number of intersections with
    the control polygon is greater than or equal to
    the number of intersections with the curve

Bezier Curve
89
Variation Diminishing
  • For any line, the number of intersections with
    the control polygon is greater than or equal to
    the number of intersections with the curve

Bezier Curve
90
Variation Diminishing
  • For any line, the number of intersections with
    the control polygon is greater than or equal to
    the number of intersections with the curve

Bezier Curve
91
Variation Diminishing
  • For any line, the number of intersections with
    the control polygon is greater than or equal to
    the number of intersections with the curve

Bezier Curve
92
Variation Diminishing
  • For any line, the number of intersections with
    the control polygon is greater than or equal to
    the number of intersections with the curve

Bezier Curve
93
Variation Diminishing
  • For any line, the number of intersections with
    the control polygon is greater than or equal to
    the number of intersections with the curve

Lagrange Interpolation
94
Applications Intersection
  • Given two Bezier curves, determine if and where
    they intersect

95
Applications Intersection
  • Check if convex hulls intersect
  • If not, return no intersection
  • If both convex hulls can be approximated with a
    straight line, intersect lines and return
    intersection
  • Otherwise subdivide and recur on subdivided pieces

96
Applications Intersection
97
Applications Intersection
98
Applications Intersection
99
Applications Intersection
100
Applications Intersection
101
Applications Intersection
102
Applications Intersection
103
Applications Intersection
104
Applications Intersection
105
Applications Intersection
106
Applications Intersection
107
Application Font Rendering
108
Application Font Rendering
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