Dr. Scott Schaefer

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Bezier Triangles andMulti-Sided Patches

• Dr. Scott Schaefer

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Triangular Patches

• How do we build triangular patches instead of
  quads?

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Triangular Patches

• How do we build triangular patches instead of
  quads?

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Triangular Patches

• How do we build triangular patches instead of
  quads?

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Triangular Patches

• How do we build triangular patches instead of
  quads?

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

• Control points pijk defined in triangular array

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deCasteljau Algorithm for Bezier Triangles

• Evaluate at (s,t,u) where stu1

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deCasteljau Algorithm for Bezier Triangles

• Evaluate at (s,t,u) where stu1

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deCasteljau Algorithm for Bezier Triangles

• Evaluate at (s,t,u) where stu1

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deCasteljau Algorithm for Bezier Triangles

• Evaluate at (s,t,u) where stu1

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deCasteljau Algorithm for Bezier Triangles

• Evaluate at (s,t,u) where stu1

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deCasteljau Algorithm for Bezier Triangles

• Evaluate at (s,t,u) where stu1

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deCasteljau Algorithm for Bezier Triangles

• Evaluate at (s,t,u) where stu1

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deCasteljau Algorithm for Bezier Triangles

• Evaluate at (s,t,u) where stu1

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deCasteljau Algorithm for Bezier Triangles

• Evaluate at (s,t,u) where stu1

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Properties of Bezier Triangles

• Convex hull

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Properties of Bezier Triangles

• Convex hull
• Boundaries are Bezier curves

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Properties of Bezier Triangles

• Convex hull
• Boundaries are Bezier curves

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Properties of Bezier Triangles

• Convex hull
• Boundaries are Bezier curves

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Properties of Bezier Triangles

• Convex hull
• Boundaries are Bezier curves

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Properties of Bezier Triangles

• Convex hull
• Boundaries are Bezier curves

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Properties of Bezier Triangles

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

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Subdividing Bezier Triangles
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Subdividing Bezier Triangles
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Subdividing Bezier Triangles
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Subdividing Bezier Triangles
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Subdividing Bezier Triangles
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Subdividing Bezier Triangles
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Subdividing Bezier Triangles
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Subdividing Bezier Triangles

• Split along longest edge

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Subdividing Bezier Triangles

• Split along longest edge

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Derivatives of Bezier Triangles
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Derivatives of Bezier Triangles
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Derivatives of Bezier Triangles
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Derivatives of Bezier Triangles
Really only 2 directions for derivatives!!!
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Continuity Between Bezier Triangles

• How do we determine continuity conditions between
  Bezier triangles?

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Continuity Between Bezier Triangles

• How do we determine continuity conditions between
  Bezier triangles?

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Continuity Between Bezier Triangles

• How do we determine continuity conditions between
  Bezier triangles?

Control points on boundary align for C0
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Continuity Between Bezier Triangles

• How do we determine continuity conditions between
  Bezier triangles?

What about C1?
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Continuity Between Bezier Triangles

• Use subdivision in parametric space!!!

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Continuity Between Bezier Triangles

• Use subdivision in parametric space!!!

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

• C1 continuity

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Continuity Between Bezier Triangles

• C1 continuity

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Continuity Between Bezier Triangles

• C1 continuity

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Multi-Sided Patches

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

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Control Points for Multi-Sided Patches
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Control Points for Multi-Sided Patches
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Control Points for Multi-Sided Patches

• Minkowski summations for multi-sided patches

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Control Points for Multi-Sided Patches

• Minkowski summations for multi-sided patches

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Control Points for Multi-Sided Patches

• Five sided control points

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Control Points for Multi-Sided Patches

• Five sided control points

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Control Points for Multi-Sided Patches

• Five sided control points

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Control Points for Multi-Sided Patches

• Five sided control points

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Control Points for Multi-Sided Patches

• Five sided control points

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Control Points for Multi-Sided Patches

• Five sided control points

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S-Patch Evaluation

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

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S-Patch Evaluation

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

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S-Patch Evaluation

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

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S-Patch Evaluation

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

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S-Patch Evaluation

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

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S-Patch Evaluation

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

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S-Patch Evaluation

• Apply barycentric coordinates to each shape in
  hierarchy

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S-Patch Evaluation

• Apply barycentric coordinates to each shape in
  hierarchy

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S-Patch Evaluation

• Apply barycentric coordinates to each shape in
  hierarchy

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S-Patch Evaluation

• Apply barycentric coordinates to each shape in
  hierarchy

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S-Patch Evaluation

• Apply barycentric coordinates to each shape in
  hierarchy

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S-Patch Evaluation

• Apply barycentric coordinates to each shape in
  hierarchy

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S-Patch Evaluation

• Apply barycentric coordinates to each shape in
  hierarchy

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S-Patch Evaluation

• Apply barycentric coordinates to each shape in
  hierarchy

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S-Patch Evaluation

• Apply barycentric coordinates to each shape in
  hierarchy

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S-Patch Properties

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

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S-Patch Oddities

• Multiple ways of defining multi-sided grids