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

2. 2/108 Problems with Interpolation

3. 3/108 Problems with Interpolation

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

5. 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. 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. 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. 8/108 Properties of Bernstein Polynomials

9. 9/108 Properties of Bernstein Polynomials

10. 10/108 Properties of Bernstein Polynomials

11. 11/108 Properties of Bernstein Polynomials Binomial Theorem:

12. 12/108 Properties of Bernstein Polynomials Binomial Theorem:

13. 13/108 Properties of Bernstein Polynomials Binomial Theorem:

14. 14/108 Properties of Bernstein Polynomials Binomial Theorem:

15. 15/108 Properties of Bernstein Polynomials

16. 16/108 Properties of Bernstein Polynomials

17. 17/108 Properties of Bernstein Polynomials

18. 18/108 Properties of Bernstein Polynomials

19. 19/108 More Properties of Bernstein Polynomials

20. 20/108 More Properties of Bernstein Polynomials

21. 21/108 More Properties of Bernstein Polynomials

22. 22/108 More Properties of Bernstein Polynomials

23. 23/108 Properties of Bernstein Polynomials

24. 24/108 Properties of Bernstein Polynomials Base case:

25. 25/108 Properties of Bernstein Polynomials Base case:

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

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

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

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

30. 30/108 Bezier Curves

31. 31/108 Bezier Curves

32. 32/108 Bezier Curves

33. 33/108 Bezier Curve Properties Interpolate end-points

34. 34/108 Bezier Curve Properties Interpolate end-points

35. 35/108 Bezier Curve Properties Interpolate end-points

36. 36/108 Bezier Curve Properties Interpolate end-points

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

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

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

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

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

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

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

44. 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. 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. 46/108 Matrix Form of Bezier Curves

47. 47/108 Matrix Form of Bezier Curves

48. 48/108 Matrix Form of Bezier Curves

49. 49/108 Matrix Form of Bezier Curves

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

51. 51/108 Change of Basis Bezier coefficients

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

53. 53/108 Change of Basis

54. 54/108 Change of Basis

55. 55/108 Change of Basis

56. 56/108 Change of Basis

57. 57/108 Change of Basis monomial coefficients

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

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

60. 60/108 Degree Elevation

61. 61/108 Degree Elevation

62. 62/108 Degree Elevation

63. 63/108 Degree Elevation

64. 64/108 Degree Elevation

65. 65/108 Degree Elevation

66. 66/108 Degree Elevation

67. 67/108 Degree Elevation

68. 68/108 Degree Elevation

69. 69/108 Degree Elevation

70. 70/108 Degree Elevation

71. 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. 72/108 Pyramid Algorithms for Bezier Curves

73. 73/108 Pyramid Algorithms for Bezier Curves

74. 74/108 Pyramid Algorithms for Bezier Curves

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

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

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

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

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

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

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

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

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

84. 84/108 Subdividing Bezier Curves

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

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

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

88. 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. 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. 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. 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. 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. 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. 94/108 Applications: Intersection Given two Bezier curves, determine if and where they intersect

95. 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. 96/108 Applications: Intersection

97. 97/108 Applications: Intersection

98. 98/108 Applications: Intersection

99. 99/108 Applications: Intersection

100. 100/108 Applications: Intersection

101. 101/108 Applications: Intersection

102. 102/108 Applications: Intersection

103. 103/108 Applications: Intersection

104. 104/108 Applications: Intersection

105. 105/108 Applications: Intersection

106. 106/108 Applications: Intersection

107. 107/108 Application: Font Rendering

108. 108/108 Application: Font Rendering