1. Computer Graphics Representing Curves and Surfaces

2. Review eq(11.5)

3. Review eq(11.6/11.7)

4. Review eq(11.8)

6. Review eq(11.9)

7. Review eq(11.10)

8. Review Blending function (also called ‘Basis’ function)

9. Hermite Curves 以曲線端點 P 1.P 4 以及端點斜率 R 1.R 4 求曲線方程式

10. Hermite Curves eq(11.12)

11. Hermite Curves eq(11.13)

12. Hermite Curves eq(11.14)

13. Hermite Curves eq(11.15)

14. Hermite Curves

15. Hermite Curves eq(11.16)

16. Hermite Curves eq(11.17)

17. Hermite Curves eq(11.18)

18. Hermite Curves eq(11.19)

19. Hermite Curves eq(11.20)

20. Hermite Curves eq(11.21)

21. Hermite Curve

22. Hermite Curve eq(11.22)

23. Hermite Curve eq(11.23) Reduce 6 multiplies and 3 additions to 3 multiplies and 3 additions.

24. Bezier Curves 以曲線端點 P 1.P 4 以及控制點 P 2.P 3 求曲線方程式 曲線端點斜率為

25. Bezier Curves eq(11.24)

26. Bezier Curves eq(11.25)

27. Bezier Curves eq(11.26)

28. Bezier Curves eq(11.27/11.28)

29. Bezier Curves eq(11.29)

30. Bezier Curves

31. Define n as the order of Bezier curves. Define i as control point.

32. Bezier Curves

34. De Casteljau iterations

35. Bezier Curves Linear Bezier splines Control points: P 0, P 1

36. Bezier Curves Quadratic Bezier splines Control points: P 0, P 1, P 2

37. Bezier Curves Quadratic Bezier splines Control points: P 0, P 1, P 2

38. Bezier Curves Cubic Bezier splines Control points: P 0, P 1, P 2, P 3

39. Bezier Curves Cubic Bezier splines Control points: P 0, P 1, P 2, P 3

40. Bezier Curves Cubic Bezier splines Control points: P 0, P 1, P 2, P 3

41. Bezier Curves

42. http://www.ibiblio.org/e-notes/Splines/Bezier.htm

43. Spline Natural cubic spline C 0, C 1, C 2 continuous. Interpolates(passes through) the control points. Moving any one control point affects the entire curve.

44. Spline B-spline Local control. Moving a control point affects only a small part of a curve. Do not interpolate their control points. Sharing control points between segments.

45. B-spline m+1 control points P 0, …, P m, m≥3 m-2 curve segments Q 3, Q 4, …, Q m For each i≥4, there is a join point or knot between Q i-1 and Q i at the parameter value t i.

46. B-spline eq(11.43/11.44)

47. B-spline eq(11.44)

48. Uniform Nonrational B-spline ‘Uniform’ means that the knots are spaced at equal intervals of the parameter t. ‘Nonrational’ is used to distinguish these splines from rational cubic polynomial curves, see Section 11.2.5 We assume that t 3 =0 and the interval t i+1 -t i =1

49. Uniform Nonrational B-spline eq(11.32/11.33/11.34)

50. Uniform Nonrational B-spline eq(11.35)

51. Uniform Nonrational B-spline eq(11.36)

52. Uniform Nonrational B-spline eq(11.37)

53. Uniform Nonrational B-spline eq(11.38)

54. Uniform Nonrational B-spline

56. Uniform Nonrational B-spline eq(11.39/11.40/11.41)

57. Uniform Nonrational B-spline eq(11.42) The curve can be forced to be interpolate specific points by replicating control points.

58. Nonuniform Nonrational B-spline Parameter interval between successive knot values need not be uniform. Blending functions are no longer the same for each interval. Continuity at selected join points can be reduced from C 2 to C 1 to C 0 to none. When the curve is C 0, the curve interpolates a control point.

59. Nonuniform Nonrational B-spline If the continuity is reduced to C 0, then the curve interpolates a control point, but without the undesirable effect of uniform B-splines, where the curve segments on either side of the interpolated control point are straight lines.

60. Nonuniform Nonrational B-spline m+1 control points P 0, P 1, …, P m nondecreasing sequence of knot values t 0, t 1, …, t m+4

61. Nonuniform Nonrational B-spline

62. Nonuniform Rational Cubic Polynomial Curve

63. Advantages They are invariant under rotation, scaling, translation and perspective transformations of the control points. They can define precisely and of the conic sections.

64. Catmull-Rom spline

66. Catmull-Rom spline eq(11.47)

67. Subdividing Curves Bezier Curve

68. Subdividing Curves

69. Uniform B-spline Curve Doubling: Do k subdivision steps (for k-degree B-spline)

70. Subdividing Curves

71. Conversion eq(11.56)

72. Conversion eq(11.57/11.58)

73. Drawing Curves eq(11.59/11.60/11.61)

74. Drawing Curves eq(11.62/11.63)

75. Drawing Curves eq(11.64/11.65/11.66/11.67)

76. Drawing Curves eq(11.68/11.69/11.70)

77. Drawing Curves eq(11.71/11.72)

78. Comparison

79. Parametric Bicubic Surfaces eq(11.73)

80. Parametric Bicubic Surfaces

81. Parametric Bicubic Surfaces eq(11.74/11.75)

82. Parametric Bicubic Surfaces eq(11.76)

83. Hermite Surfaces eq(11.77)

84. Hermite Surfaces

85. Hermite Surfaces eq(11.78)

86. Hermite Surfaces eq(11.79/11.80/11.81)

87. Hermite Surfaces eq(11.82/11.83)

88. Hermite Surfaces eq(11.84)

89. Hermite Surfaces

90. Normals to Surfaces eq(11.88/11.89)

91. Normals to Surfaces eq(11.90)