1. 1 Dr. Scott Schaefer Tensor-Product Surfaces

2. 2/64 Smooth Surfaces Lagrange Surfaces  Interpolating sets of curves Bezier Surfaces B-spline Surfaces

3. 3/64 Lagrange Surfaces

4. 4/64 Lagrange Surfaces

5. 5/64 Lagrange Surfaces

6. 6/64 Lagrange Surfaces

7. 7/64 Lagrange Surfaces

8. 8/64 Lagrange Surfaces

9. 9/64 Lagrange Surfaces

10. 10/64 Lagrange Surfaces

11. 11/64 Lagrange Surfaces

12. 12/64 Lagrange Surfaces

13. 13/64 Lagrange Surfaces

14. 14/64 Lagrange Surfaces

15. 15/64 Lagrange Surfaces

16. 16/64 Lagrange Surfaces

17. 17/64 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. 18/64 Interpolating Sets of Curves Given a set of parametric curves p 0 (t), p 1 (t), …, p n (t), build a surface that interpolates them

19. 19/64 Interpolating Sets of Curves Given a set of parametric curves p 0 (t), p 1 (t), …, p n (t), build a surface that interpolates them Evaluate each curve at parameter value t, then use these points as the control points for a Lagrange curve of degree n Evaluate this new curve at parameter value s

20. 20/64 Bezier Surfaces

21. 21/64 Bezier Surfaces

22. 22/64 Bezier Surfaces

23. 23/64 Bezier Surfaces

24. 24/64 Bezier Surfaces

25. 25/64 Bezier Surfaces

26. 26/64 Bezier Surfaces

27. 27/64 Bezier Surfaces

28. 28/64 Bezier Surfaces

29. 29/64 Bezier Surfaces

30. 30/64 Bezier Surfaces

31. 31/64 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. 32/64 General Tensor Product Surfaces

33. 33/64 General Tensor Product Surfaces

34. 34/64 Properties Curve properties/algorithms apply to surfaces too  Convex hull

35. 35/64 Properties Curve properties/algorithms apply to surfaces too  Convex hull  Degree elevation

36. 36/64 Properties Curve properties/algorithms apply to surfaces too  Convex hull  Degree elevation  Evaluation algorithms

37. 37/64 Properties Curve properties/algorithms apply to surfaces too  Convex hull  Degree elevation  Evaluation algorithms  ….  Analog of variation diminishing does not apply!!!

38. 38/64 Matrix Form of Quadrilateral Bezier Patches

39. 39/64 Matrix Form of Quadrilateral Bezier Patches

40. 40/64 deCasteljau Algorithm for Bezier Surfaces

41. 41/64 deCasteljau Algorithm for Bezier Surfaces

42. 42/64 deCasteljau Algorithm for Bezier Surfaces

43. 43/64 deCasteljau Algorithm for Bezier Surfaces

44. 44/64 deCasteljau Algorithm for Bezier Surfaces

45. 45/64 deCasteljau Algorithm for Bezier Surfaces

46. 46/64 deCasteljau Algorithm for Bezier Surfaces

47. 47/64 deCasteljau Algorithm for Bezier Surfaces

48. 48/64 deCasteljau Algorithm for Bezier Surfaces

49. 49/64 Derivatives of Bezier Surfaces Exact evaluate in the s-direction and use those control points to compute derivative in t-direction Exact evaluate in the t-direction and use those control points to compute derivative in s-direction Use a pyramid algorithm to compute derivatives

50. 50/64 Derivatives using deCasteljau’s algorithm

51. 51/64 Derivatives using deCasteljau’s algorithm

52. 52/64 Derivatives using deCasteljau’s algorithm

53. 53/64 Blossoming for Tensor-Product Patches Symmetry: b(s 1,s 2,…,s m |t 1,t 2,…,t n ) = b(s q(1),s q(2),…,s q(m) |t r(1),t r(2),…,t r(n) ) for any permutation q of (1,…,m) and r of (1,...,n) Multi-affine: b(s 1,…,(1-d)s k +d v k,,…s m |t 1,…,(1-e)t j +e w j,,…t n ) = (1-d)(1-e) b(s 1,…,s k,,…s m |t 1,…,t j,,…t n ) + (1-d)e b(s 1,…,s k,,…s m |t 1,…,w j,,…t n ) + de b(s 1,…,v k,,…s m |t 1,…,w j,,…t n ) + d(1-e) b(s 1,…,v k,,…s m |t 1,…,t j,,…t n ) Diagonal: b(s,s,…,s|t,t,…,t) = p(s,t)

54. 54/64 Curves on Bezier Surfaces Assume we have points p(s 1,t 1 ), p(s 2,t 2 ) on a Bezier surface p(s,t). Construct a curve on the Bezier surface between these two points.

55. 55/64 Curves on Bezier Surfaces Assume we have points p(s 1,t 1 ), p(s 2,t 2 ) on a Bezier surface p(s,t). Construct a curve on the Bezier surface between these two points.

56. 56/64 Curves on Bezier Surfaces Assume we have points p(s 1,t 1 ), p(s 2,t 2 ) on a Bezier surface p(s,t). Construct a curve on the Bezier surface between these two points.

57. 57/64 Curves on Bezier Surfaces Assume we have points p(s 1,t 1 ), p(s 2,t 2 ) on a Bezier surface p(s,t). Construct a curve on the Bezier surface between these two points.

58. 58/64 Curves on Bezier Surfaces Assume we have points p(s 1,t 1 ), p(s 2,t 2 ) on a Bezier surface p(s,t). Construct a curve on the Bezier surface between these two points.

59. 59/64 Curves on Bezier Surfaces Assume we have points p(s 1,t 1 ), p(s 2,t 2 ) on a Bezier surface p(s,t). Construct a curve on the Bezier surface between these two points.

60. 60/64 Curves on Bezier Surfaces Assume we have points p(s 1,t 1 ), p(s 2,t 2 ) on a Bezier surface p(s,t). Construct a curve on the Bezier surface between these two points. k th control point for Bezier curve of degree n+m!!!

61. 61/64 Triangular Patches How do we build triangular patches instead of quads?

62. 62/64 Triangular Patches How do we build triangular patches instead of quads?

63. 63/64 Triangular Patches How do we build triangular patches instead of quads?

64. 64/64 Triangular Patches How do we build triangular patches instead of quads? Continuity difficult to maintain between patches Parameterization very distorted Not symmetric