1. 1 Dr. Scott Schaefer Subdivision Curves

2. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as output Recursively applying S yields a sequence of curves which should converge to some limit shape

3. 3/96 Subdivision Rules Typically chosen to be linear combinations of neighboring vertices Rules usually depend only on local topology of shape Reposition Old Vertices Insert New Vertices

4. 4/96 Applying Subdivision Rules

5. 5/96 Applying Subdivision Rules

6. 6/96 Applying Subdivision Rules

7. 7/96 Applying Subdivision Rules

8. 8/96 Applying Subdivision Rules

9. 9/96 Applying Subdivision Rules

10. 10/96 Applying Subdivision Rules

11. 11/96 Applying Subdivision Rules

12. 12/96 Applying Subdivision Rules

13. 13/96 Applying Subdivision Rules

14. 14/96 Applying Subdivision Rules

15. 15/96 Applying Subdivision Rules

16. 16/96 Subdivision Rules Via Blossoming Assume knot-spacing uniform 0,2,4,6,8,… Find control points for refined knots 0,1,2,3,…

17. 17/96 Subdivision Rules Via Blossoming Assume knot-spacing uniform 0,2,4,6,8,… Find control points for refined knots 0,1,2,3,… GivenFind

18. 18/96 Subdivision Rules Via Blossoming Assume knot-spacing uniform 0,2,4,6,8,… Find control points for refined knots 0,1,2,3,… GivenFind

19. 19/96 Subdivision Rules Via Blossoming Assume knot-spacing uniform 0,2,4,6,8,… Find control points for refined knots 0,1,2,3,… GivenFind

20. 20/96 Subdivision Rules Via Blossoming Assume knot-spacing uniform 0,2,4,6,8,… Find control points for refined knots 0,1,2,3,… GivenFind Works for arbitrary degree B-splines!!!

21. 21/96 Lane Reisenfeld Subdivision Linearly subdivide the curve by inserting the midpoint on each edge Perform averaging by replacing each edge by its midpoint d times

22. 22/96 Lane Reisenfeld Subdivision

23. 23/96 Lane Reisenfeld Subdivision

24. 24/96 Lane Reisenfeld Subdivision

25. 25/96 Lane Reisenfeld Subdivision

26. 26/96 Lane Reisenfeld Subdivision

27. 27/96 Lane Reisenfeld Subdivision

28. 28/96 Lane Reisenfeld Subdivision

29. 29/96 Lane Reisenfeld Subdivision

30. 30/96 Lane Reisenfeld Subdivision

31. 31/96 Interpolatory Subdivision Interpolating control vertices may be desirable Catmull-Rom splines are not refinable!!! Reposition Old Vertices Insert New Vertices

32. 32/96 Four-Point Subdivision

33. 33/96 Four-Point Subdivision

34. 34/96 Four-Point Subdivision

35. 35/96 Four-Point Subdivision

36. 36/96 Four-Point Subdivision

37. 37/96 Four-Point Subdivision

38. 38/96 Four-Point Subdivision

39. 39/96 Four-Point Subdivision

40. 40/96 Four-Point Subdivision

41. 41/96 Four-Point Subdivision

42. 42/96 Four-Point Subdivision

43. 43/96 Four-Point Subdivision

44. 44/96 Four-Point Subdivision

45. 45/96 Four-Point Subdivision

46. 46/96 Four-Point Subdivision

47. 47/96 Four-Point Subdivision

48. 48/96 Four-Point Subdivision

49. 49/96 Subdivision as Basis Function Refinement

50. 50/96 Subdivision as Basis Function Refinement Cubic B-spline Basis Function

51. 51/96 Subdivision as Basis Function Refinement Four-Point Basis Function

52. 52/96 Subdivision as Basis Function Refinement

53. 53/96 Subdivision as Basis Function Refinement

54. 54/96 Subdivision as Basis Function Refinement

55. 55/96 Subdivision as Basis Function Refinement

56. 56/96 Subdivision as Basis Function Refinement

57. 57/96 Subdivision as Basis Function Refinement

58. 58/96 Subdivision as Basis Function Refinement

59. 59/96 Subdivision as Basis Function Refinement

60. 60/96 Limit Points of Curve Subdivision

61. 61/96 Limit Points of Curve Subdivision

62. 62/96 Limit Points of Curve Subdivision

63. 63/96 Limit Points of Curve Subdivision

64. 64/96 Limit Points of Curve Subdivision

65. 65/96 Limit Points of Curve Subdivision

66. 66/96 Limit Points of Curve Subdivision Symmetry!!!

67. 67/96 Limit Points of Curve Subdivision

68. 68/96 Limit Points of Curve Subdivision Zero!!!

69. 69/96 Limit Points of Curve Subdivision

70. 70/96 Limit Points of Curve Subdivision

71. 71/96 Limit Points of Curve Subdivision

72. 72/96 Limit Points of Curve Subdivision

73. 73/96 Limit Points of Curve Subdivision

74. 74/96 Limit Points of Curve Subdivision

75. 75/96 Limit Points of Curve Subdivision

76. 76/96 Evaluate scaling relationship at Solve linear system of equations with constraint Limit Points of Curve Subdivision

77. 77/96 Limit Points of Curve Subdivision

78. 78/96 Limit Points of Curve Subdivision Assume N(x) has finite support Let y be the smallest parameter such that N(y)=0 and for all x>y N(x)=0

79. 79/96 Limit Points of Curve Subdivision Assume N(x) has finite support Let y be the smallest parameter such that N(y)=0 and for all x>y N(x)=0

80. 80/96 Limit Points of Curve Subdivision Assume N(x) has finite support Let y be the smallest parameter such that N(y)=0 and for all x>y N(x)=0

81. 81/96 Limit Points of Curve Subdivision Assume N(x) has finite support Let y be the smallest parameter such that N(y)=0 and for all x>y N(x)=0

82. 82/96 Limit Points of Curve Subdivision

83. 83/96 Limit Points of Curve Subdivision

84. 84/96 Limit Points of Curve Subdivision

85. 85/96 Limit Points of Curve Subdivision Limit mask is left-eigenvector corresponding to 1

86. 86/96 Limit Points of Curve Subdivision

87. 87/96 Limit Points of Curve Subdivision

88. 88/96 Derivatives of Subdivision Curves

89. 89/96 Derivatives of Subdivision Curves

90. 90/96 Derivatives of Subdivision Curves

91. 91/96 Derivatives of Subdivision Curves left-eigenvector corresponding to 1/2

92. 92/96 Derivatives of Subdivision Curves

93. 93/96 Four-Point Limits and Derivatives

94. 94/96 Four-Point Limits and Derivatives

95. 95/96 Four-Point Limits and Derivatives

96. 96/96 Four-Point Limits and Derivatives