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).


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




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)





Constructing Tangents to Points on a Parabola
Constructing Tangents to Points on a Parabola

Slope of Y123 at X2.

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