Bezier Triangles and Multi-Sided Patches

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© University of Wisconsin, CS559 Spring 2004

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Presentation transcript:

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