1. 1 Dr. Scott Schaefer Bezier Triangles and Multi-Sided Patches

2. 2/72 Triangular Patches How do we build triangular patches instead of quads?

3. 3/72 Triangular Patches How do we build triangular patches instead of quads?

4. 4/72 Triangular Patches How do we build triangular patches instead of quads?

5. 5/72 Triangular Patches How do we build triangular patches instead of quads? Continuity difficult to maintain between patches Parameterization very distorted Not symmetric

6. 6/72 Bezier Triangles Control points p ijk defined in triangular array

7. 7/72 deCasteljau Algorithm for Bezier Triangles Evaluate at (s,t,u) where s+t+u=1

8. 8/72 deCasteljau Algorithm for Bezier Triangles Evaluate at (s,t,u) where s+t+u=1

9. 9/72 deCasteljau Algorithm for Bezier Triangles Evaluate at (s,t,u) where s+t+u=1

10. 10/72 deCasteljau Algorithm for Bezier Triangles Evaluate at (s,t,u) where s+t+u=1

11. 11/72 deCasteljau Algorithm for Bezier Triangles Evaluate at (s,t,u) where s+t+u=1

12. 12/72 deCasteljau Algorithm for Bezier Triangles Evaluate at (s,t,u) where s+t+u=1

13. 13/72 deCasteljau Algorithm for Bezier Triangles Evaluate at (s,t,u) where s+t+u=1

14. 14/72 deCasteljau Algorithm for Bezier Triangles Evaluate at (s,t,u) where s+t+u=1

15. 15/72 deCasteljau Algorithm for Bezier Triangles Evaluate at (s,t,u) where s+t+u=1

16. 16/72 Properties of Bezier Triangles Convex hull

17. 17/72 Properties of Bezier Triangles Convex hull Boundaries are Bezier curves

18. 18/72 Properties of Bezier Triangles Convex hull Boundaries are Bezier curves

19. 19/72 Properties of Bezier Triangles Convex hull Boundaries are Bezier curves

20. 20/72 Properties of Bezier Triangles Convex hull Boundaries are Bezier curves

21. 21/72 Properties of Bezier Triangles Convex hull Boundaries are Bezier curves

22. 22/72 Properties of Bezier Triangles Convex hull Boundaries are Bezier curves Explicit polynomial form

23. 23/72 Subdividing Bezier Triangles

24. 24/72 Subdividing Bezier Triangles

25. 25/72 Subdividing Bezier Triangles

26. 26/72 Subdividing Bezier Triangles

27. 27/72 Subdividing Bezier Triangles

28. 28/72 Subdividing Bezier Triangles

29. 29/72 Subdividing Bezier Triangles

30. 30/72 Subdividing Bezier Triangles Split along longest edge

31. 31/72 Subdividing Bezier Triangles Split along longest edge

32. 32/72 Derivatives of Bezier Triangles

33. 33/72 Derivatives of Bezier Triangles

34. 34/72 Derivatives of Bezier Triangles

35. 35/72 Derivatives of Bezier Triangles Really only 2 directions for derivatives!!!

36. 36/72 Continuity Between Bezier Triangles How do we determine continuity conditions between Bezier triangles?

37. 37/72 Continuity Between Bezier Triangles How do we determine continuity conditions between Bezier triangles?

38. 38/72 Continuity Between Bezier Triangles How do we determine continuity conditions between Bezier triangles? Control points on boundary align for C 0

39. 39/72 Continuity Between Bezier Triangles How do we determine continuity conditions between Bezier triangles? What about C 1 ?

40. 40/72 Continuity Between Bezier Triangles Use subdivision in parametric space!!!

41. 41/72 Continuity Between Bezier Triangles Use subdivision in parametric space!!! First k rows of triangles from subdivision yield C k continuity conditions

42. 42/72 Continuity Between Bezier Triangles C 1 continuity

43. 43/72 Continuity Between Bezier Triangles C 1 continuity

44. 44/72 Continuity Between Bezier Triangles C 1 continuity

45. 45/72 Multi-Sided Patches Multi-sided holes in surfaces can be difficult to fill Construct a generalized Bezier patch for multi-sided holes

46. 46/72 Control Points for Multi-Sided Patches

47. 47/72 Control Points for Multi-Sided Patches

48. 48/72 Control Points for Multi-Sided Patches Minkowski summations for multi-sided patches

49. 49/72 Control Points for Multi-Sided Patches Minkowski summations for multi-sided patches

50. 50/72 Control Points for Multi-Sided Patches Five sided control points

51. 51/72 Control Points for Multi-Sided Patches Five sided control points

52. 52/72 Control Points for Multi-Sided Patches Five sided control points

53. 53/72 Control Points for Multi-Sided Patches Five sided control points

54. 54/72 Control Points for Multi-Sided Patches Five sided control points

55. 55/72 Control Points for Multi-Sided Patches Five sided control points

56. 56/72 S-Patch Evaluation Given a point inside parametric domain, find barycentric coordinates w.r.t. convex hull of domain

57. 57/72 S-Patch Evaluation Given a point inside parametric domain, find barycentric coordinates w.r.t. convex hull of domain

58. 58/72 S-Patch Evaluation Given a point inside parametric domain, find barycentric coordinates w.r.t. convex hull of domain

59. 59/72 S-Patch Evaluation Given a point inside parametric domain, find barycentric coordinates w.r.t. convex hull of domain

60. 60/72 S-Patch Evaluation Given a point inside parametric domain, find barycentric coordinates w.r.t. convex hull of domain

61. 61/72 S-Patch Evaluation Given a point inside parametric domain, find barycentric coordinates w.r.t. convex hull of domain

62. 62/72 S-Patch Evaluation Apply barycentric coordinates to each shape in hierarchy

63. 63/72 S-Patch Evaluation Apply barycentric coordinates to each shape in hierarchy

64. 64/72 S-Patch Evaluation Apply barycentric coordinates to each shape in hierarchy

65. 65/72 S-Patch Evaluation Apply barycentric coordinates to each shape in hierarchy

66. 66/72 S-Patch Evaluation Apply barycentric coordinates to each shape in hierarchy

67. 67/72 S-Patch Evaluation Apply barycentric coordinates to each shape in hierarchy

68. 68/72 S-Patch Evaluation Apply barycentric coordinates to each shape in hierarchy

69. 69/72 S-Patch Evaluation Apply barycentric coordinates to each shape in hierarchy

70. 70/72 S-Patch Evaluation Apply barycentric coordinates to each shape in hierarchy

71. 71/72 S-Patch Properties Boundary curves are Bezier curve Convex hull Surface is rational because barycentric coordinates used are rational functions

72. 72/72 S-Patch Oddities Multiple ways of defining multi-sided grids