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Bezier Triangles and Multi-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 s+t+u=1
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deCasteljau Algorithm for Bezier Triangles
Evaluate at (s,t,u) where s+t+u=1
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deCasteljau Algorithm for Bezier Triangles
Evaluate at (s,t,u) where s+t+u=1
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deCasteljau Algorithm for Bezier Triangles
Evaluate at (s,t,u) where s+t+u=1
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deCasteljau Algorithm for Bezier Triangles
Evaluate at (s,t,u) where s+t+u=1
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deCasteljau Algorithm for Bezier Triangles
Evaluate at (s,t,u) where s+t+u=1
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deCasteljau Algorithm for Bezier Triangles
Evaluate at (s,t,u) where s+t+u=1
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deCasteljau Algorithm for Bezier Triangles
Evaluate at (s,t,u) where s+t+u=1
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deCasteljau Algorithm for Bezier Triangles
Evaluate at (s,t,u) where s+t+u=1
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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
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 Index has number of entries equal to vertices in base shape
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Control Points for Multi-Sided Patches
Five sided control points Index has number of entries equal to vertices in base shape Entries positive and sum to d
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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 Construction Start with a polygonal domain
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S-Patch Construction Given a point inside parametric domain, find barycentric coordinates w.r.t. domain
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S-Patch Construction Control points in domain are given by
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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 if barycentric coordinates used are rational functions
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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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S-Patch Oddities Multiple ways of defining multi-sided grids
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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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