Barycentric coordinates and transfinite interpolation

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