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Unformatted text preview: Barycentric Coordinates Sumant Pattanaik Barycentric Coordinates Given v find weights w i such that are barycentric coordinates v is said to have an affine combination of the vertices. Weights are unique for triangles. = i i i i i w p w v 2 p 1 p 3 p v j j i w w Barycentric Coordinates Given v find weights w i such that are barycentric coordinates Given values at , you can interpolate = i i i i i w p w v 2 p 1 p 3 p v j j i w w i f i p = i i i i i v w f w f Properties of Barycentric Coordinates Interpolates values at vertices Linear on boundary 1 f 2 f 3 f 4 f 5 f Boundary Value Interpolation Interpolates values at vertices Linear on boundary Smooth on interior 1 f 2 f 3 f 4 f 5 f Boundary Value Interpolation Boundary Value Interpolation Extend texture to interior Surface Deformation = i i i i i w p w v i p v Surface Deformation = i i i i i w p w v i p v Surface Deformation = i i i i i w p w v i p = i i i i i w p w v i p v v Surface Deformation = i i i i i w p w v v i p = i i i i i w p w v v i p Coordinates In A Polytope Points form vertices of a closed polytope v lies on the (hyper) plane of the polytope (inside or outside) Example: A 2D triangle Unique (barycentric): signed area of the triangle Can be extended to any ND simplex A general polytope Nonunique The triangle technique is not applicable. 3 2 1 v v x A w = 1 v x 2 v 3 v x Previous Work 2D Polygons Wachspress [Wachspress 75][Loop 89][Meyer 02][Malsch 04] Barycentric within convex shapes Discrete harmonic [Desbrun 02][Floater 06] Homogeneous within convex shapes Mean value [Floater 03][Hormann 06] Homogeneous within any closed shape, barycentric within convex shapes and kernels of starshapes 3D Polyhedrons and Beyond Wachspress [Warren 96][Ju 05] Discrete harmonic [Meyer 02] Mean value [Floater 05][Ju 05] Previous Work convex polygons [ Wachspress 1975 ] closed polygons [ Floater 2003, Hormann 2004 ] Previous Work convex polygons [ Wachspress 1975 ] closed polygons [ Floater 2003, Hormann 2004 ] Previous Work convex polygons [ Wachspress 1975 ] closed polygons [ Floater 2003, Hormann 2004 ] Previous Work convex polygons [ Wachspress 1975 ] closed polygons [ Floater 2003, Hormann 2004 ] Previous Work...
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 Spring '10
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