1. Smooth Surfaces
Dr. Scott Schaefer

2. Smooth Surfaces Lagrange Surfaces Interpolating sets of curves
Bezier Surfaces
B-spline Surfaces
Subdivision Surfaces

3. Lagrange Surfaces

4. Lagrange Surfaces

5. Lagrange Surfaces

6. Lagrange Surfaces

7. Lagrange Surfaces

8. Lagrange Surfaces

9. Lagrange Surfaces

10. Lagrange Surfaces

11. Lagrange Surfaces

12. Lagrange Surfaces

13. Lagrange Surfaces

14. Lagrange Surfaces

15. Lagrange Surfaces

16. Lagrange Surfaces

17. Lagrange Surfaces – Properties
Surface interpolates all control points
The boundaries of the surface are Lagrange curves defined by the control points on the boundary

18. Interpolating Sets of Curves
Given a set of parametric curves p0(u), p1(u), …, pn(u) , build a surface that interpolates them

19. Interpolating Sets of Curves
Given a set of parametric curves p0(u), p1(u), …, pn(u) , build a surface that interpolates them
Evaluate each curve at parameter value u, then use these points as the control points for a Lagrange curve of degree n
Evaluate this new curve at parameter value v

20. Bezier Surfaces

21. Bezier Surfaces

22. Bezier Surfaces

23. Bezier Surfaces

24. Bezier Surfaces

25. Bezier Surfaces

26. Bezier Surfaces

27. Bezier Surfaces

28. Bezier Surfaces

29. Bezier Surfaces

30. Bezier Surfaces

31. Bezier Surfaces – Properties
Surface lies in convex hull of control points
Surface interpolates the four corner control points
Boundary curves are Bezier
curves defined only by
control points on boundary

32. B-spline Surfaces

33. B-spline Surfaces

34. B-spline Surfaces

35. B-spline Surfaces

36. B-spline Surfaces

37. B-spline Surfaces

38. B-spline Surfaces

39. B-spline Surfaces

40. B-spline Surfaces

41. B-spline Surfaces

42. B-spline Surfaces

43. B-spline Surfaces

44. B-spline Surfaces

45. B-spline Surfaces

46. B-splines Surfaces Example

47. B-splines Surfaces Example

48. B-splines Surfaces Example

49. B-splines Surfaces Example

50. B-splines Surfaces Example

51. B-splines Surfaces Example

52. B-splines Surfaces Example

53. B-splines Surfaces Example

54. B-splines Surfaces Example

55. B-splines Surfaces Example

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

57. Arbitrary Topology Surfaces

58. Arbitrary Topology Surfaces

59. Subdivision Surfaces
Originally a generalization of B-spline surfaces to arbitrary topology
Guaranteed to be smooth
Geri’s Game copyright Pixar

60. Subdivision Surfaces
Set of rules S applied recursively to some polygon shape p0
pk+1 = S(pk)

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

62. Linear Subdivision

63. Linear Subdivision

64. Averaging

65. Averaging

66. Averaging
Centroid (average of four black vertices)

67. Averaging

68. Subdivision Surfaces – Examples

69. Subdivision Surfaces – Examples

70. Subdivision Surfaces – Examples

71. Subdivision Surfaces – Examples

72. Subdivision Surfaces – Examples

73. Subdivision Surfaces – Examples

74. Subdivision Surfaces – Examples

75. Subdivision Surfaces – Examples

76. Subdivision Surfaces – Examples

77. 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)

78. 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)]

79. 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)

80. Subdivision Surfaces – Examples

81. Subdivision Surfaces – Examples

82. Subdivision Surfaces – Examples

83. Subdivision Surfaces – Examples

84. Subdivision Surfaces – Examples

85. Subdivision Surfaces – Examples

86. Subdivision Surfaces – Examples

87. Subdivision Surfaces – Examples

88. Subdivision Surfaces – Examples

89. Subdivision Surfaces – Examples

90. Subdivision Surfaces – Examples

91. Subdivision Surfaces – Examples

92. Subdivision Surfaces – Examples

93. Subdivision Surfaces – Examples

94. Subdivision Surfaces – Examples

95. Subdivision Surfaces – Examples

96. Subdivision Surfaces – Examples

97. Subdivision Surfaces – Examples