Barycentric coordinates and transfinite interpolation

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Linear system of three equations, ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 1 1 λ 1 1 ⎝v1 1 v2 1 v3 1 ⎠ ⎝λ 2 ⎠ = ⎝x 1 ⎠ . v1 2 v2 2 v3 2 λ 3 x 2 By Cramer’s rule, unique solution is λ 1 = A 1 A , λ 2 = A 2 A , λ 3 = A 3 A . v 3 A A 2 x A 3 A 1 v 1 v 2
Properties ◮ Lagrange property: λ i (v j ) = δ ij . ◮ Interpolation: if g(x) = 3∑ λ i (x)f (v i ), i=1 then g(v i ) = f (v i ). ◮ Linear precision: if f is linear then g = f . ◮ Linearity: λ i is linear. ◮ Bernstein-Bezier basis for polynomials, degree d: p(x) = ∑ i+j+k=d n! i!j!k! λi 1(x)λ j 2 (x)λk 3(x)c ijk .
Page 1 and 2: Barycentric coordinates and transfi
Page 3: Barycentric coordinates on a triang
Page 7 and 8: Properties ◮ Lagrange property:
Page 9 and 10: To find λ and µ: let a = v 00 −
Page 11 and 12: Proof: Meyer et al. 2002 Let A i =
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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