# Using the Aitken Construction to Evaluate Pointwise Tangents to a Parabola

Starting for the Lagrange Three Point Polynomial I will develop a construction that can be used to evaluate the Slopes of the Parabola at its defining points.

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 parabola that passes through the points (X1,Y1), (X2,Y2) and (X3,Y3) is then

Y123=FLINE(X1,Y12,X3,Y23,X)

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

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

The results for the tangents at the points (X1,Y1), (X2,Y2) and (X3,Y3) can be expressedin the following way.

Slope at (X1,Y1) =( Y12@3-Y13@2)/(X3-X2)

Slope at (X2,Y2)=(Y12@3-Y23@1)/(X3-X1)

Slope at (X3,Y3)=(Y13@2-Y23@1)/(X2-X1)

Where:

Y12@3=FLINE(X1,Y1,X2,Y2,X3)

Y23@1=FLINE(X2,Y2,X3,Y3,X1)

Y13@2=FLINE(X1,Y1,X3,Y3,X2) Constructing Tangents to Points on a Parabola

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