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 tn+1
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 aren’t pretty
Is there an easier way to evaluate the equation of a Bezier curve?

72. Pyramid Algorithms for Bezier Curves

73. Pyramid Algorithms for Bezier Curves

74. Pyramid Algorithms for Bezier Curves

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

76. Pyramid Algorithms for Derivatives 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

109. Application: Vector Graphics