1. Subdivision Surfaces
Dr. Scott Schaefer

2. B-spline Surfaces

3. B-spline Surfaces

4. B-spline Surfaces

5. B-spline Surfaces

6. B-spline Surfaces

7. B-spline Surfaces

8. B-spline Surfaces

9. B-spline Surfaces

10. B-spline Surfaces

11. B-spline Surfaces

12. B-spline Surfaces

13. B-spline Surfaces

14. B-spline Surfaces

15. B-spline Surfaces Example

16. B-spline Surfaces Example

17. B-spline Surfaces Example

18. B-spline Surfaces Example

19. B-spline Surfaces Example

20. B-spline Surfaces Example

21. B-spline Surfaces Example

22. B-spline Surfaces Example

23. B-spline Surfaces Example

24. B-spline Surfaces Example

25. B-spline Surface – Properties
Surface inside convex hull of control points
Guaranteed to be smooth everywhere
Smoothness is determined by number of averaging steps

26. Arbitrary Topology Surfaces

27. Arbitrary Topology Surfaces

28. Subdivision Surfaces Used in movie and game industries
Supported by most 3D modeling software
Toy Story © Disney / Pixar
Geri’s Game © Pixar Animation Studios

29. Subdivision Surfaces Set of rules S that recursively act on a shape
Arbitrary topology surfaces
Smooth everywhere

30. Repeated Averaging Assume surface is made out of quads
Any number of quads may touch a single vertex
Subdivision rules: linear subdivision followed by averaging

31. Linear Subdivision

32. Linear Subdivision

33. Averaging

34. Averaging

35. Averaging

36. Subdivision Surfaces – Examples

37. Subdivision Surfaces – Examples

38. Subdivision Surfaces – Examples

39. Subdivision Surfaces – Examples

40. Subdivision Surfaces – Examples

41. Subdivision Surfaces – Examples

42. Subdivision Surfaces – Examples

43. Subdivision Surfaces – Examples

44. Subdivision Surfaces – Examples

45. Implementing Linear Subdivision
linearSub ( F, V )
newV = V
newF = {}
for each face Fi
for j = 1 to 4
ej = getVert ( Fi,j, Fi,j+1)
add centroid to newV and store index in c
add face (Fi,j, ej, c, ej-1) to newF
return (newF, newV)

46. Implementing Linear Subdivision
getVert ( i1, i2 )
if orderless key (i1,i2) not in hash
add midpoint of V[i1], V[i2] to newV
hash[(i1,i2)] = index of new point
return hash[(i1,i2)]

47. Implementing Averaging
Average( F, V )
newV = 0 * V
val = array of 0 whose size is number of vertices
newF = F
for each face Fi
cent = centroid for Fi
newV[Fi] += cent
val[Fi] += 1
for each vertex newV[i]
newV[i] /= val[i]
return (newF, newV)

48. Subdivision Surfaces – Examples

49. Subdivision Surfaces – Examples

50. Subdivision Surfaces – Examples

51. Subdivision Surfaces – Examples

52. Subdivision Surfaces – Examples

53. Subdivision Surfaces – Examples

54. Subdivision Surfaces – Examples

55. Subdivision Surfaces – Examples

56. Subdivision Surfaces – Examples

57. Subdivision Surfaces – Examples

58. Subdivision Surfaces – Examples

59. Subdivision Surfaces – Examples

60. Subdivision Surfaces – Examples

61. Subdivision Surfaces – Examples

62. Subdivision Surfaces – Examples

63. Subdivision Surfaces – Examples

64. Subdivision Surfaces – Examples

65. Subdivision Surfaces – Examples

66. Catmull-Clark Subdivision

67. Catmull-Clark Subdivision

68. Catmull-Clark Subdivision

69. Catmull-Clark Subdivision

70. Catmull-Clark Subdivision

71. Catmull-Clark Subdivision

72. Catmull-Clark Subdivision
midpoint

73. Catmull-Clark Subdivision

74. Catmull-Clark

75. Catmull-Clark Subdivision

76. Catmull-Clark Subdivision

77. Catmull-Clark Subdivision
Repeated Averaging
Catmull-Clark

78. Catmull-Clark Subdivision
One round of subdivision produces all quads
C2 almost everywhere
C1 at vertices with valence 4
Most commonly used subdivision scheme in existence

79. Doo-Sabin Subdivision

80. Doo-Sabin Subdivision

81. Doo-Sabin Subdivision

82. Doo-Sabin Subdivision

83. Doo-Sabin Subdivision

84. Doo-Sabin Subdivision
Variant of repeated averaging subdivision
All vertices valence 4 after one round of subdivision
C1 everywhere

85. Midedge Subdivision

86. Midedge Subdivision

87. Midedge Subdivision

88. Midedge Subdivision
Two rounds of midedge subdivision similar to one round of Doo-Sabin

89. Midedge Subdivision
Simplest subdivision scheme in terms of rules (midpoint)
Produces C1 surfaces everywhere
Very slow convergence at high valence vertices and fast convergence
at low valence vertices creates
artifacts

90. Loop Subdivision

91. Loop Subdivision

92. Loop Subdivision

93. Loop Subdivision

94. Loop Subdivision

95. Loop Subdivision

96. Loop Subdivision Operates only on triangle surfaces
Produces C2 surfaces almost everywhere
C1 at vertices with valence 6

97. Sqrt(3) Subdivision

98. Sqrt(3) Subdivision

99. Sqrt(3) Subdivision

100. Sqrt(3) Subdivision

101. Sqrt(3) Subdivision
Two rounds of sqrt(3) subdivision yield a ternary split of the original surface!!!

102. Sqrt(3) Subdivision

103. Sqrt(3) Subdivision

104. Sqrt(3) Subdivision

105. Sqrt(3) Subdivision
Produces all triangles after one round of subdivision
Creates C2 surfaces almost everywhere
C1 at vertices with valence 6

106. Interpolatory Quadrilateral Subdivision
Tensor-Product of four-point curve subdivision

107. Interpolatory Quadrilateral Subdivision
Tensor-Product of four-point curve subdivision

108. Interpolatory Quadrilateral Subdivision
Tensor-Product of four-point curve subdivision

109. Interpolatory Quadrilateral Subdivision
Tensor-Product of four-point curve subdivision

110. Interpolatory Quadrilateral Subdivision
Tensor-Product of four-point curve subdivision

111. Interpolatory Quadrilateral Subdivision
Tensor-Product of four-point curve subdivision
Same result!!!

112. Interpolatory Quadrilateral Subdivision
Tensor-Product of four-point curve subdivision

113. Interpolatory Quadrilateral Subdivision
Tensor-Product of four-point curve subdivision

114. Interpolatory Quadrilateral Subdivision
Tensor-Product of four-point curve subdivision

115. Interpolatory Quadrilateral Subdivision
Tensor-Product of four-point curve subdivision
Same result!!!

116. Interpolatory Quadrilateral Subdivision
Tensor-Product of four-point curve subdivision

117. Interpolatory Quadrilateral Subdivision
Construct a virtual point v such that tensor-product is the same

118. Interpolatory Quadrilateral Subdivision
Construct a virtual point v such that tensor-product is the same

119. Interpolatory Quadrilateral Subdivision
Based on four-point curve subdivision
Interpolates vertices
Operates only on quadrilateral surfaces
Creates C1 surfaces everywhere

120. Butterfly Subdivision

121. Butterfly Subdivision

122. Butterfly Subdivision

123. Butterfly Subdivision

124. Butterfly Subdivision

125. Butterfly Subdivision

126. Butterfly Subdivision

127. Butterfly Subdivision
Interpolatory subdivision scheme for triangle meshes
Produces surfaces that are C1 everywhere

128. Comparisons Loop Butterfly Catmull-Clark Doo-Sabin
Image taken from “Subdivision for Modeling and Animation” Siggraph 2000 Course Notes by Zorin et al.

129. Comparisons Loop Butterfly Catmull-Clark Doo-Sabin
Image taken from “Subdivision for Modeling and Animation” Siggraph 2000 Course Notes by Zorin et al.

130. Catmull-Clark (triangulated)
Comparisons
Mesh
Loop (triangulated)
Catmull-Clark
Catmull-Clark (triangulated)
Image taken from “Subdivision for Modeling and Animation” Siggraph 2000 Course Notes by Zorin et al.