1 Dr. Scott Schaefer The Bernstein Basis and Bezier Curves

2/108 Problems with Interpolation

3/108 Problems with Interpolation

4/108 Bezier Curves Polynomial curves that seek to approximate rather than to interpolate

5/108 Bernstein Polynomials Degree 1: (1-t), t Degree 2: (1-t) 2, 2(1-t)t, t 2 Degree 3: (1-t) 3, 3(1-t) 2 t, 3(1-t)t 2, t 3

6/108 Bernstein Polynomials Degree 1: (1-t), t Degree 2: (1-t) 2, 2(1-t)t, t 2 Degree 3: (1-t) 3, 3(1-t) 2 t, 3(1-t)t 2, t 3 Degree 4: (1-t) 4, 4(1-t) 3 t, 6(1-t) 2 t 2, 4(1-t)t 3,t 4

7/108 Bernstein Polynomials Degree 1: (1-t), t Degree 2: (1-t) 2, 2(1-t)t, t 2 Degree 3: (1-t) 3, 3(1-t) 2 t, 3(1-t)t 2, t 3 Degree 4: (1-t) 4, 4(1-t) 3 t, 6(1-t) 2 t 2, 4(1-t)t 3,t 4 Degree 5: (1-t) 5, 5(1-t) 4 t, 10(1-t) 3 t 2, 10(1-t) 2 t 3,5(1-t)t 4,t 5 … Degree n: for

8/108 Properties of Bernstein Polynomials

9/108 Properties of Bernstein Polynomials

10/108 Properties of Bernstein Polynomials

11/108 Properties of Bernstein Polynomials Binomial Theorem:

12/108 Properties of Bernstein Polynomials Binomial Theorem:

13/108 Properties of Bernstein Polynomials Binomial Theorem:

14/108 Properties of Bernstein Polynomials Binomial Theorem:

15/108 Properties of Bernstein Polynomials

16/108 Properties of Bernstein Polynomials

17/108 Properties of Bernstein Polynomials

18/108 Properties of Bernstein Polynomials

19/108 More Properties of Bernstein Polynomials

20/108 More Properties of Bernstein Polynomials

21/108 More Properties of Bernstein Polynomials

22/108 More Properties of Bernstein Polynomials

23/108 Properties of Bernstein Polynomials

24/108 Properties of Bernstein Polynomials Base case:

25/108 Properties of Bernstein Polynomials Base case:

26/108 Properties of Bernstein Polynomials Base case: Inductive Step: Assume

27/108 Properties of Bernstein Polynomials Base case: Inductive Step: Assume

28/108 Properties of Bernstein Polynomials Base case: Inductive Step: Assume

29/108 Properties of Bernstein Polynomials Base case: Inductive Step: Assume

30/108 Bezier Curves

31/108 Bezier Curves

32/108 Bezier Curves

33/108 Bezier Curve Properties Interpolate end-points

34/108 Bezier Curve Properties Interpolate end-points

35/108 Bezier Curve Properties Interpolate end-points

36/108 Bezier Curve Properties Interpolate end-points

37/108 Bezier Curve Properties Interpolate end-points Tangent at end-points in direction of first/last edge

38/108 Bezier Curve Properties Interpolate end-points Tangent at end-points in direction of first/last edge

39/108 Bezier Curve Properties Interpolate end-points Tangent at end-points in direction of first/last edge

40/108 Bezier Curve Properties Interpolate end-points Tangent at end-points in direction of first/last edge

41/108 Bezier Curve Properties Interpolate end-points Tangent at end-points in direction of first/last edge

42/108 Bezier Curve Properties Interpolate end-points Tangent at end-points in direction of first/last edge Another Bezier curve of vectors!!!

43/108 Bezier Curve Properties Interpolate end-points Tangent at end-points in direction of first/last edge

44/108 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

45/108 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

46/108 Matrix Form of Bezier Curves

47/108 Matrix Form of Bezier Curves

48/108 Matrix Form of Bezier Curves

49/108 Matrix Form of Bezier Curves

50/108 Matrix Form of Bezier Curves Computation in monomial basis is unstable!!! Most proofs/computations are easier in Bernstein basis!!!

51/108 Change of Basis Bezier coefficients

52/108 Change of Basis Coefficients in monomial basis!!! Bezier coefficients

53/108 Change of Basis

54/108 Change of Basis

55/108 Change of Basis

56/108 Change of Basis

57/108 Change of Basis monomial coefficients

58/108 Change of Basis monomial coefficients Coefficients in Bezier basis!!!

59/108 Degree Elevation Power basis is trivial: add 0 t n+1 What about Bezier basis?

60/108 Degree Elevation

61/108 Degree Elevation

62/108 Degree Elevation

63/108 Degree Elevation

64/108 Degree Elevation

65/108 Degree Elevation

66/108 Degree Elevation

67/108 Degree Elevation

68/108 Degree Elevation

69/108 Degree Elevation

70/108 Degree Elevation

71/108 Pyramid Algorithms for Bezier Curves Polynomials aren’t pretty Is there an easier way to evaluate the equation of a Bezier curve?

72/108 Pyramid Algorithms for Bezier Curves

73/108 Pyramid Algorithms for Bezier Curves

74/108 Pyramid Algorithms for Bezier Curves

75/108 Pyramid Algorithms for Derivatives of Bezier Curves Take derivative of any level of pyramid!!! (up to constant multiple)

76/108 Pyramid Algorithms for Derivatives of Bezier Curves Take derivative of any level of pyramid!!! (up to constant multiple)

77/108 Subdividing Bezier Curves Given a single Bezier curve, construct two smaller Bezier curves whose union is exactly the original curve

78/108 Subdividing Bezier Curves Given a single Bezier curve, construct two smaller Bezier curves whose union is exactly the original curve

79/108 Subdividing Bezier Curves Given a single Bezier curve, construct two smaller Bezier curves whose union is exactly the original curve

80/108 Subdividing Bezier Curves Given a single Bezier curve, construct two smaller Bezier curves whose union is exactly the original curve

81/108 Subdividing Bezier Curves Given a single Bezier curve, construct two smaller Bezier curves whose union is exactly the original curve

82/108 Subdividing Bezier Curves Control points for left curve!!!

83/108 Subdividing Bezier Curves Control points for right curve!!!

84/108 Subdividing Bezier Curves

85/108 Variation Diminishing Intuitively means that the curve “wiggles” no more than its control polygon

86/108 Variation Diminishing Intuitively means that the curve “wiggles” no more than its control polygon Lagrange Interpolation

87/108 Variation Diminishing Intuitively means that the curve “wiggles” no more than its control polygon Bezier Curve

88/108 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

89/108 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

90/108 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

91/108 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

92/108 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

93/108 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

94/108 Applications: Intersection Given two Bezier curves, determine if and where they intersect

95/108 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

96/108 Applications: Intersection

97/108 Applications: Intersection

98/108 Applications: Intersection

99/108 Applications: Intersection

100/108 Applications: Intersection

101/108 Applications: Intersection

102/108 Applications: Intersection

103/108 Applications: Intersection

104/108 Applications: Intersection

105/108 Applications: Intersection

106/108 Applications: Intersection

107/108 Application: Font Rendering

108/108 Application: Font Rendering