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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

16/72 Properties of Bezier Triangles Convex hull

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

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

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

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

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

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

23/72 Subdividing Bezier Triangles

24/72 Subdividing Bezier Triangles

25/72 Subdividing Bezier Triangles

26/72 Subdividing Bezier Triangles

27/72 Subdividing Bezier Triangles

28/72 Subdividing Bezier Triangles

29/72 Subdividing Bezier Triangles

30/72 Subdividing Bezier Triangles Split along longest edge

31/72 Subdividing Bezier Triangles Split along longest edge

32/72 Derivatives of Bezier Triangles

33/72 Derivatives of Bezier Triangles

34/72 Derivatives of Bezier Triangles

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

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

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

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

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

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

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

42/72 Continuity Between Bezier Triangles C 1 continuity

43/72 Continuity Between Bezier Triangles C 1 continuity

44/72 Continuity Between Bezier Triangles C 1 continuity

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

46/72 Control Points for Multi-Sided Patches

47/72 Control Points for Multi-Sided Patches

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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