Barycentric coordinates and transfinite interpolation
Barycentric coordinates and transfinite interpolation
Barycentric coordinates and transfinite interpolation
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To use this construction we need to find ∂ψ<br />
∂n (y) <strong>and</strong> ∂g 0<br />
∂n (y).<br />
Theorem<br />
If d(M E , ∂Ω) > 0 <strong>and</strong> d(M I , ∂Ω) > 0 <strong>and</strong> y ∈ ∂Ω then<br />
∂ψ<br />
∂n (y) = 1 ,<br />
V n−1<br />
∂g 0<br />
∂n (y) = 1 f (p(y, v)) − f (y)<br />
V n−1<br />
∫H(n) ‖p(y, v) − y‖<br />
dv,<br />
where V n−1 is the volume of the unit sphere in R n−1 : V 1 = 2,<br />
V 2 = π, V 3 = 4π/3, V 4 = π 2 /2,...
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