Bezier Triangles and MultiSided Patches Dr Scott Schaefer
Bezier Triangles and Multi-Sided Patches Dr. Scott Schaefer 1
Triangular Patches n How do we build triangular patches instead of quads? 2/67
Triangular Patches n How do we build triangular patches instead of quads? 3/67
Triangular Patches n How do we build triangular patches instead of quads? 4/67
Triangular Patches n How do we build triangular patches instead of quads? Continuity difficult to maintain between patches Parameterization very distorted Not symmetric 5/67
Bezier Triangles n Control points pijk defined in triangular array 6/67
de. Casteljau Algorithm for Bezier Triangles n Evaluate at (s, t, u) where s+t+u=1 7/67
de. Casteljau Algorithm for Bezier Triangles n Evaluate at (s, t, u) where s+t+u=1 8/67
de. Casteljau Algorithm for Bezier Triangles n Evaluate at (s, t, u) where s+t+u=1 9/67
de. Casteljau Algorithm for Bezier Triangles n Evaluate at (s, t, u) where s+t+u=1 10/67
de. Casteljau Algorithm for Bezier Triangles n Evaluate at (s, t, u) where s+t+u=1 11/67
de. Casteljau Algorithm for Bezier Triangles n Evaluate at (s, t, u) where s+t+u=1 12/67
de. Casteljau Algorithm for Bezier Triangles n Evaluate at (s, t, u) where s+t+u=1 13/67
de. Casteljau Algorithm for Bezier Triangles n Evaluate at (s, t, u) where s+t+u=1 14/67
de. Casteljau Algorithm for Bezier Triangles n Evaluate at (s, t, u) where s+t+u=1 15/67
Properties of Bezier Triangles n Convex hull 16/67
Properties of Bezier Triangles Convex hull n Boundaries are Bezier curves n 17/67
Properties of Bezier Triangles Convex hull n Boundaries are Bezier curves n 18/67
Properties of Bezier Triangles Convex hull n Boundaries are Bezier curves n 19/67
Properties of Bezier Triangles Convex hull n Boundaries are Bezier curves n 20/67
Properties of Bezier Triangles Convex hull n Boundaries are Bezier curves n 21/67
Properties of Bezier Triangles Convex hull n Boundaries are Bezier curves n Explicit polynomial form n 22/67
Subdividing Bezier Triangles 23/67
Subdividing Bezier Triangles 24/67
Subdividing Bezier Triangles 25/67
Subdividing Bezier Triangles 26/67
Subdividing Bezier Triangles 27/67
Subdividing Bezier Triangles 28/67
Subdividing Bezier Triangles 29/67
Subdividing Bezier Triangles n Split alongest edge 30/67
Subdividing Bezier Triangles n Split alongest edge 31/67
Derivatives of Bezier Triangles 32/67
Derivatives of Bezier Triangles 33/67
Derivatives of Bezier Triangles 34/67
Derivatives of Bezier Triangles Really only 2 directions for derivatives!!! 35/67
Continuity Between Bezier Triangles n How do we determine continuity conditions between Bezier triangles? 36/67
Continuity Between Bezier Triangles n How do we determine continuity conditions between Bezier triangles? 37/67
Continuity Between Bezier Triangles n How do we determine continuity conditions between Bezier triangles? Control points on boundary align for C 0 38/67
Continuity Between Bezier Triangles n How do we determine continuity conditions between Bezier triangles? What about C 1? 39/67
Continuity Between Bezier Triangles n Use subdivision in parametric space!!! 40/67
Continuity Between Bezier Triangles n Use subdivision in parametric space!!! First k rows of triangles from subdivision yield Ck continuity conditions 41/67
Continuity Between Bezier Triangles n C 1 continuity 42/67
Continuity Between Bezier Triangles n C 1 continuity 43/67
Continuity Between Bezier Triangles n C 1 continuity 44/67
Multi-Sided Patches Multi-sided holes in surfaces can be difficult to fill n Construct a generalized Bezier patch for multi-sided holes n 45/67
Control Points for Multi-Sided Patches n Five sided control points 46/67
Control Points for Multi-Sided Patches n Five sided control points 47/67
Control Points for Multi-Sided Patches n Five sided control points Index has number of entries equal to vertices in base shape 48/67
Control Points for Multi-Sided Patches n Five sided control points Index has number of entries equal to vertices in base shape Entries positive and sum to d 49/67
Control Points for Multi-Sided Patches n Five sided control points 50/67
Control Points for Multi-Sided Patches n Five sided control points 51/67
Control Points for Multi-Sided Patches n Five sided control points 52/67
Control Points for Multi-Sided Patches n Five sided control points 53/67
Control Points for Multi-Sided Patches n Five sided control points 54/67
S-Patch Construction n Start with a polygonal domain 55/67
S-Patch Construction n Given a point inside parametric domain, find barycentric coordinates w. r. t. domain 56/67
S-Patch Construction n Control points in domain are given by 57/67
S-Patch Evaluation n Apply barycentric coordinates to each shape in hierarchy 58/67
S-Patch Evaluation n Apply barycentric coordinates to each shape in hierarchy 59/67
S-Patch Evaluation n Apply barycentric coordinates to each shape in hierarchy 60/67
S-Patch Evaluation n Apply barycentric coordinates to each shape in hierarchy 61/67
S-Patch Evaluation n Apply barycentric coordinates to each shape in hierarchy 62/67
S-Patch Evaluation n Apply barycentric coordinates to each shape in hierarchy 63/67
S-Patch Evaluation n Apply barycentric coordinates to each shape in hierarchy 64/67
S-Patch Evaluation n Apply barycentric coordinates to each shape in hierarchy 65/67
S-Patch Evaluation n Apply barycentric coordinates to each shape in hierarchy 66/67
S-Patch Properties Boundary curves are Bezier curve n Convex hull n Surface is rational if barycentric coordinates used are rational functions n 67/67
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Control Points for Multi-Sided Patches 69/67
Control Points for Multi-Sided Patches 70/67
Control Points for Multi-Sided Patches n Minkowski summations for multi-sided patches 71/67
Control Points for Multi-Sided Patches n Minkowski summations for multi-sided patches 72/67
S-Patch Oddities n Multiple ways of defining multi-sided grids 73/67
S-Patch Evaluation n Given a point inside parametric domain, find barycentric coordinates w. r. t. convex hull of domain 74/67
S-Patch Evaluation n Given a point inside parametric domain, find barycentric coordinates w. r. t. convex hull of domain 75/67
S-Patch Evaluation n Given a point inside parametric domain, find barycentric coordinates w. r. t. convex hull of domain 76/67
S-Patch Evaluation n Given a point inside parametric domain, find barycentric coordinates w. r. t. convex hull of domain 77/67
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