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Simple Asymptotic Estimates for the Perimeter of an Ellipse Using the Ivory Series Representation

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Abstract

For the perimeter P(a, b) of an ellipse with the semi-axes \(a\ge b>0\), a sequence \(P_n(a,b)\) is constructed such that the relative error of the approximation \( P(a,b)\approx P_n(a,b)\) satisfies the following inequalities

$$\begin{aligned} 0\le \frac{P(a,b)-P_n(a,b)}{P(a,b)}\le \frac{3}{\pi (2n+1)^3}\left( \frac{a-b}{a+b}\right) ^{2n+2}\quad (n\in {\mathbb {N}}). \end{aligned}$$

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Notes

  1. We have \(\epsilon =\sqrt{1-q^2}\), where \(q:=b/a\) is called the aspect ratio of an ellipse.

  2. In geodetic science called the third flattening.

  3. \(\prod _{j=m}^n x_j:=1\), for \(m>n\); consequently \(w_0=1\)

  4. All the graphics and calculations in this paper are made using the Mathematica [9] computer system.

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  1. Faculty of Civil and Geodetic Engineering, University of Ljubljana, 1000, Ljubljana 1, Slovenia

    Vito Lampret

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  1. Vito Lampret
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Lampret, V. Simple Asymptotic Estimates for the Perimeter of an Ellipse Using the Ivory Series Representation. Mediterr. J. Math. 18, 52 (2021). https://doi.org/10.1007/s00009-020-01650-z

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  • DOI: https://doi.org/10.1007/s00009-020-01650-z

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