Bezier Triangles and Multi-Sided Patches Dr. Scott Schaefer
Triangular Patches How do we build triangular patches instead of quads?
Triangular Patches How do we build triangular patches instead of quads?
Triangular Patches How do we build triangular patches instead of quads?
Triangular Patches How do we build triangular patches instead of quads? Parameterization very distorted Continuity difficult to maintain between patches Not symmetric
Bezier Triangles Control points pijk defined in triangular array
deCasteljau Algorithm for Bezier Triangles Evaluate at (s,t,u) where s+t+u=1
deCasteljau Algorithm for Bezier Triangles Evaluate at (s,t,u) where s+t+u=1
deCasteljau Algorithm for Bezier Triangles Evaluate at (s,t,u) where s+t+u=1
deCasteljau Algorithm for Bezier Triangles Evaluate at (s,t,u) where s+t+u=1
deCasteljau Algorithm for Bezier Triangles Evaluate at (s,t,u) where s+t+u=1
deCasteljau Algorithm for Bezier Triangles Evaluate at (s,t,u) where s+t+u=1
deCasteljau Algorithm for Bezier Triangles Evaluate at (s,t,u) where s+t+u=1
deCasteljau Algorithm for Bezier Triangles Evaluate at (s,t,u) where s+t+u=1
deCasteljau Algorithm for Bezier Triangles Evaluate at (s,t,u) where s+t+u=1
Properties of Bezier Triangles Convex hull
Properties of Bezier Triangles Convex hull Boundaries are Bezier curves
Properties of Bezier Triangles Convex hull Boundaries are Bezier curves
Properties of Bezier Triangles Convex hull Boundaries are Bezier curves
Properties of Bezier Triangles Convex hull Boundaries are Bezier curves
Properties of Bezier Triangles Convex hull Boundaries are Bezier curves
Properties of Bezier Triangles Convex hull Boundaries are Bezier curves Explicit polynomial form
Subdividing Bezier Triangles
Subdividing Bezier Triangles
Subdividing Bezier Triangles
Subdividing Bezier Triangles
Subdividing Bezier Triangles
Subdividing Bezier Triangles
Subdividing Bezier Triangles
Subdividing Bezier Triangles Split along longest edge
Subdividing Bezier Triangles Split along longest edge
Derivatives of Bezier Triangles
Derivatives of Bezier Triangles
Derivatives of Bezier Triangles
Derivatives of Bezier Triangles Really only 2 directions for derivatives!!!
Continuity Between Bezier Triangles How do we determine continuity conditions between Bezier triangles?
Continuity Between Bezier Triangles How do we determine continuity conditions between Bezier triangles?
Continuity Between Bezier Triangles How do we determine continuity conditions between Bezier triangles? Control points on boundary align for C0
Continuity Between Bezier Triangles How do we determine continuity conditions between Bezier triangles? What about C1?
Continuity Between Bezier Triangles Use subdivision in parametric space!!!
Continuity Between Bezier Triangles Use subdivision in parametric space!!! First k rows of triangles from subdivision yield Ck continuity conditions
Continuity Between Bezier Triangles C1 continuity
Continuity Between Bezier Triangles C1 continuity
Continuity Between Bezier Triangles C1 continuity
Multi-Sided Patches Multi-sided holes in surfaces can be difficult to fill Construct a generalized Bezier patch for multi-sided holes
Control Points for Multi-Sided Patches Five sided control points
Control Points for Multi-Sided Patches Five sided control points
Control Points for Multi-Sided Patches Five sided control points Index has number of entries equal to vertices in base shape
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
Control Points for Multi-Sided Patches Five sided control points
Control Points for Multi-Sided Patches Five sided control points
Control Points for Multi-Sided Patches Five sided control points
Control Points for Multi-Sided Patches Five sided control points
Control Points for Multi-Sided Patches Five sided control points
S-Patch Construction Start with a polygonal domain
S-Patch Construction Given a point inside parametric domain, find barycentric coordinates w.r.t. domain
S-Patch Construction Control points in domain are given by
S-Patch Evaluation Apply barycentric coordinates to each shape in hierarchy
S-Patch Evaluation Apply barycentric coordinates to each shape in hierarchy
S-Patch Evaluation Apply barycentric coordinates to each shape in hierarchy
S-Patch Evaluation Apply barycentric coordinates to each shape in hierarchy
S-Patch Evaluation Apply barycentric coordinates to each shape in hierarchy
S-Patch Evaluation Apply barycentric coordinates to each shape in hierarchy
S-Patch Evaluation Apply barycentric coordinates to each shape in hierarchy
S-Patch Evaluation Apply barycentric coordinates to each shape in hierarchy
S-Patch Evaluation Apply barycentric coordinates to each shape in hierarchy
S-Patch Properties Boundary curves are Bezier curve Convex hull Surface is rational if barycentric coordinates used are rational functions
Control Points for Multi-Sided Patches
Control Points for Multi-Sided Patches
Control Points for Multi-Sided Patches Minkowski summations for multi-sided patches
Control Points for Multi-Sided Patches Minkowski summations for multi-sided patches
S-Patch Oddities Multiple ways of defining multi-sided grids
S-Patch Evaluation Given a point inside parametric domain, find barycentric coordinates w.r.t. convex hull of domain
S-Patch Evaluation Given a point inside parametric domain, find barycentric coordinates w.r.t. convex hull of domain
S-Patch Evaluation Given a point inside parametric domain, find barycentric coordinates w.r.t. convex hull of domain
S-Patch Evaluation Given a point inside parametric domain, find barycentric coordinates w.r.t. convex hull of domain