Aitken’s Graphical Construction for a Parabola

This paper presents a detailed demonstration of the exact nature of Aitken’s construction for generating points on a parabola.

It also introduces the FLINE, a function for generating points on a line, providing a direct way to encode Aitken’s construction.

It will be shown that points on a parabola generated by Aitken’s construction are identical to the points generated by a second order Lagrange polynomial.

Additional work shows how to construct tangents on the Parabola.

Please note: MathPix, GreenShot, Notepad++ proved to be extremely helpful in composing this paper.

Aitken, A. (1932). On Interpolation by Iteration of Proportional Parts, without the Use of Differences. 

Proceedings of the Edinburgh Mathematical Society, 3(1), 56-76. doi:10.1017/S0013091500013808

Introducing FLINE to evaluate points on the line defined by end points (X1, Y1) and (X2, Y2),

it is then easy to encode Aitken’s Construction

Expanding and collecting like terms:

reproduces the second order Lagrange polynomial.

A Side note:

Generating a point using Lagrange polynomial requires 3 divisions, 15 multiplications, 6 subtractions and 3 additions.

On the other hand, Aitken formulation only requires 3 divisions, 6 multiplications, 9 subtractions and 3 additions.

Extension of the Construction to Evaluate the Tangents at Defining Points of the Parabola

Using FLINE as a point of focus. Further examination of the Aitken Construction led to the realization that local tangents to the parabola could also be constructed.

The development of the construction will begin with the polynomial. Taking the first derivative of Lagrange Polynomial

and evaluating it at a particular point say (X2, Y2)


The Tangent at point (X2, Y2) can then be express in terms of FLINE

Continuing, all three tangents can then be evaluated

It should be noted that:

Showing how the tangent can be computed from the secants.

SLOPE: Numerical Differentiation of Tabular Functions


This report presents a unique approach for the Numerical Differentiation of Tabular Functions. This approach focuses on the local shape of the Tabular Function, selecting all sets of three adjacent data points. The key features of this process are represented in the following diagrams.

Consider a typical interval of the Tabular function. Focus on point J, where the Data Point J runs from Point 2 to  Point N-1. At each interval construct, the Secants related to the adjacent points.

The Slope at Point J is estimated by the average of the Tangent Estimates that fall between the   MIN(SL, SR) and the MAX(SL, SR).

The notion of employing averaging as a means to stabilize the slope computation was inspired by various descriptions of Numerical Integration techniques that use sequential averaging to produce a smooth integral, suppressing irregularities in Tabular Functions.

Tangent Estimators

With the three points that make up each interval of the Tabular Function, it is possible to develop local fits to the data based on conic sections.

The complete formula for a conic curve is the classical formula:

Recognizing that the conic segment models must include the terms to support translation in both the X and Y directions (the slope of the curve is invariant under translation), gives rise to the choice of seven conic segments that can be defined by three points:

The detailed derivation of the central differences provided by the tangent estimators is given in a supporting document.

A Collection of Tangent Estimates for Numerical Differentiation.pdf

Evaluating Tangent Estimators

The equations for all the Tangent Estimators are folded into a single subroutine. Computing numerators and denominators separately provide a simple control method to avoid the issue of division by zero.

Slope Estimates for the First and Last Data Points

The slopes of the first and last data points are computed using an extrapolation based on the Aitken construction for a Parabola.

Details of the Aitken Construction can be found here:

Aitken Graphical Constructions for Points on a Parabola.pdf

A Collection of Tangent Estimates for Numerical Differentiation

The purpose of this report is to presents a collection of Tangent Estimates that can be used for the Numerical Differentiation of Tabular Functions. Derived by fitting Conic Segments to three adjacent points on the Tabular Function, these central difference formulas have been reduced to equations expressed in terms of the local secants.

Tangent Estimate Formulas

Secant Identities that simplified the Tangent Estimate Formulas.

Example development of the Tangent Estimate for the X Parabola.

Note: All Equations were typeset using MathPix