Implicit Hermitian Interpolation based on a Modification to the Four Point Aitken Cubic Construction

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:

Y2233=FLINE(X2,Y123,X3,Y234,X)

Y123=FLINE(X1,Y12,X2,Y23,X)

Y234=FLINE(X2,Y23,X4,Y34,X)

Y12=FLINE(X1,Y1,X2,Y2,X)

Y23=FLINE(X2,Y2,X3,Y3,X)

I contend that this is a construction for the Hermetian interpolation that goes exactly through points (X2,Y2) and (X3,Y3).

Y2233 atX2

and is tangent to the Parabola Y123 at point (X2,Y2) and tangent to the Parabola Y234 at point (X3,Y3)

Slope of Y2233 at X2

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