Aitken’s Four Point Construction is Equivalent to the Four Point Lagrange Polynomial

Using the FLINE equations for a straight line, I will show that the Four Point Aitken Cubic Construction is equivalent to the Four Point Lagrange Polynomial for a Cubic.

The FLINE form provides a compact expression for the parametric equation for the straight line from the left-most point (XL,YL) to the right-most point (XR,YR).

Y=FLINE(XL,YL,XR,YR,X)=(YL(XR-X)+YR(X-XL))/(XR-XL)

Using this form the Aitken Construction for points on the cubic curve that passes through the points (X1,Y1), (X2,Y2), (X3,Y3) and (X4,Y4) is then:

Y1234=FLINE(X1,Y123,X4,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)

Equivalence_Cubic