While exploring the Four Point Aitken Construction for the Cubic
the final step in the process:
Y1234 = FLINE(X1,Y123,X4,Y234,X)
I made the following graphically inspired modification:
I contend that this is a construction for the Hermetian interpolation that goes exactly through points (X2,Y2) and (X3,Y3).
and is tangent to the Parabola Y123 at point (X2,Y2) and tangent to the Parabola Y234 at point (X3,Y3)