Dr' Scott Schaefer

1
Bezier Triangles andMulti-Sided Patches

• Dr. Scott Schaefer

2
Triangular Patches

• How do we build triangular patches instead of
  quads?

3
Triangular Patches

• How do we build triangular patches instead of
  quads?

4
Triangular Patches

• How do we build triangular patches instead of
  quads?

5
Triangular Patches

• How do we build triangular patches instead of
  quads?

Parameterization very distorted
Continuity difficult to maintain between patches
Not symmetric
6
Bezier Triangles

• Control points pijk defined in triangular array

7
deCasteljau Algorithm for Bezier Triangles

• Evaluate at (s,t,u) where stu1

8
deCasteljau Algorithm for Bezier Triangles

• Evaluate at (s,t,u) where stu1

9
deCasteljau Algorithm for Bezier Triangles

• Evaluate at (s,t,u) where stu1

10
deCasteljau Algorithm for Bezier Triangles

• Evaluate at (s,t,u) where stu1

11
deCasteljau Algorithm for Bezier Triangles

• Evaluate at (s,t,u) where stu1

12
deCasteljau Algorithm for Bezier Triangles

• Evaluate at (s,t,u) where stu1

13
deCasteljau Algorithm for Bezier Triangles

• Evaluate at (s,t,u) where stu1

14
deCasteljau Algorithm for Bezier Triangles

• Evaluate at (s,t,u) where stu1

15
deCasteljau Algorithm for Bezier Triangles

• Evaluate at (s,t,u) where stu1

16
Properties of Bezier Triangles

• Convex hull

17
Properties of Bezier Triangles

• Convex hull
• Boundaries are Bezier curves

18
Properties of Bezier Triangles

• Convex hull
• Boundaries are Bezier curves

19
Properties of Bezier Triangles

• Convex hull
• Boundaries are Bezier curves

20
Properties of Bezier Triangles

• Convex hull
• Boundaries are Bezier curves

21
Properties of Bezier Triangles

• Convex hull
• Boundaries are Bezier curves

22
Properties of Bezier Triangles

• Convex hull
• Boundaries are Bezier curves
• Explicit polynomial form

23
Subdividing Bezier Triangles
24
Subdividing Bezier Triangles
25
Subdividing Bezier Triangles
26
Subdividing Bezier Triangles
27
Subdividing Bezier Triangles
28
Subdividing Bezier Triangles
29
Subdividing Bezier Triangles
30
Subdividing Bezier Triangles

• Split along longest edge

31
Subdividing Bezier Triangles

• Split along longest edge

32
Derivatives of Bezier Triangles
33
Derivatives of Bezier Triangles
34
Derivatives of Bezier Triangles
35
Derivatives of Bezier Triangles
Really only 2 directions for derivatives!!!
36
Continuity Between Bezier Triangles

• How do we determine continuity conditions between
  Bezier triangles?

37
Continuity Between Bezier Triangles

• How do we determine continuity conditions between
  Bezier triangles?

38
Continuity Between Bezier Triangles

• How do we determine continuity conditions between
  Bezier triangles?

Control points on boundary align for C0
39
Continuity Between Bezier Triangles

• How do we determine continuity conditions between
  Bezier triangles?

What about C1?
40
Continuity Between Bezier Triangles

• Use subdivision in parametric space!!!

41
Continuity Between Bezier Triangles

• Use subdivision in parametric space!!!

First k rows of triangles from subdivision yield
Ck continuity conditions
42
Continuity Between Bezier Triangles

• C1 continuity

43
Continuity Between Bezier Triangles

• C1 continuity

44
Continuity Between Bezier Triangles

• C1 continuity

45
Multi-Sided Patches

• Multi-sided holes in surfaces
• can be difficult to fill
• Construct a generalized
• Bezier patch for multi-sided
• holes

46
Control Points for Multi-Sided Patches
47
Control Points for Multi-Sided Patches
48
Control Points for Multi-Sided Patches

• Minkowski summations for multi-sided patches

49
Control Points for Multi-Sided Patches

• Minkowski summations for multi-sided patches

50
Control Points for Multi-Sided Patches

• Five sided control points

51
Control Points for Multi-Sided Patches

• Five sided control points

52
Control Points for Multi-Sided Patches

• Five sided control points

53
Control Points for Multi-Sided Patches

• Five sided control points

54
Control Points for Multi-Sided Patches

• Five sided control points

55
Control Points for Multi-Sided Patches

• Five sided control points

56
S-Patch Evaluation

• Given a point inside parametric domain, find
  barycentric coordinates w.r.t. convex hull of
  domain

57
S-Patch Evaluation

• Given a point inside parametric domain, find
  barycentric coordinates w.r.t. convex hull of
  domain

58
S-Patch Evaluation

• Given a point inside parametric domain, find
  barycentric coordinates w.r.t. convex hull of
  domain

59
S-Patch Evaluation

• Given a point inside parametric domain, find
  barycentric coordinates w.r.t. convex hull of
  domain

60
S-Patch Evaluation

• Given a point inside parametric domain, find
  barycentric coordinates w.r.t. convex hull of
  domain

61
S-Patch Evaluation

• Given a point inside parametric domain, find
  barycentric coordinates w.r.t. convex hull of
  domain

62
S-Patch Evaluation

• Apply barycentric coordinates to each shape in
  hierarchy

63
S-Patch Evaluation

• Apply barycentric coordinates to each shape in
  hierarchy

64
S-Patch Evaluation

• Apply barycentric coordinates to each shape in
  hierarchy

65
S-Patch Evaluation

• Apply barycentric coordinates to each shape in
  hierarchy

66
S-Patch Evaluation

• Apply barycentric coordinates to each shape in
  hierarchy

67
S-Patch Evaluation

• Apply barycentric coordinates to each shape in
  hierarchy

68
S-Patch Evaluation

• Apply barycentric coordinates to each shape in
  hierarchy

69
S-Patch Evaluation

• Apply barycentric coordinates to each shape in
  hierarchy

70
S-Patch Evaluation

• Apply barycentric coordinates to each shape in
  hierarchy

71
S-Patch Properties

• Boundary curves are Bezier curve
• Convex hull
• Surface is rational because
• barycentric coordinates used
• are rational functions