Barycentric coordinates and transfinite interpolation

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Wachspress coordinates: Wachspress 1973, Warren 1996 v i+1 x v i v i−1 For a convex polygon, n vertices, let λ i (x) = w i (x) ∑ n j=1 w j(x) , w i(x) = Then λ 1 , . . . , λ n are barycentric coordinates. A(v i−1 , v i , v i+1 ) A(x, v i−1 , v i )A(x, v i , v i+1 ) .
Proof: Meyer et al. 2002 Let A i = A(x, v i , v i+1 ) and B i = A(v i−1 , v i , v i+1 ). Then express x as and rearrange: x = A i v i−1 + (B i − A i−1 − A i ) v i + A i−1 v i+1 , B i B i B i B i (v i − x) = 1 (v i − v i−1 ) − 1 (v i+1 − v i ). A i−1 A i A i−1 A i Summing both sides over i gives ∑ B i (v i − x) = 0 A i−1 A i or i ∑ w i (x)(v i − x) = 0. i
Page 1 and 2: Barycentric coordinates and transfi
Page 3 and 4: Barycentric coordinates on a triang
Page 5 and 6: Properties ◮ Lagrange property:
Page 7 and 8: Properties ◮ Lagrange property:
Page 9: To find λ and µ: let a = v 00 −
Page 13 and 14: Application: barycentric mappings R
Page 15 and 16: Theorem (F. & Kosinka). Wachspress
Page 17 and 18: Proof: With e i = (v i − x)/‖v
Page 19 and 20: Transfinite interpolation, Warren e
Page 21 and 22: Examples of MV interpolants
Page 23 and 24: On arbitrary curves p ( x, θ) 2 p
Page 25 and 26: Application: mesh deformation, Ju,
Page 27 and 28: Method 1: weight function The cubic
Page 29 and 30: To use this construction we need to
Page 31 and 32: Method 2: radial minimization MV in
Page 33 and 34: Example of radial minimization

Barycentric coordinates and transfinite interpolation

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