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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 ˆ ˆ i p ˆ ∑ ∑ = i i i i i w p w v i p v v ˆ Surface Deformation ∑ ∑ = i i i i i w p w v v i p ∑ ∑ = i i i i i w p w v ˆ ˆ v ˆ i p ˆ 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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 Polyhedron, WI, Barycentric Coordinates, Mean Value Coordinates

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