Barycentric coordinates and transfinite interpolation

Method 1: weight function
The cubic interpolant p to the data f (0), f ′ (0), f (1), f ′ (1), can be
expressed as
p(x) = g 0 (x) + ψ(x)g 1 (x),
where
g 0 (x) = (1 − x)f (0) + xf (1),
ψ(x) = x(1 − x),
g 1 (x) = (1 − x)m 0 + xm 1 ,
and
m 0 = f ′ (0) − f (1) + f (0), m 1 = −f ′ (1) + f (1) − f (0).