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A Refined Symmetric Mean Integral Approach to Bounding the Perimeter of an Ellipse

29 April 2025, Version 2
Working Paper
This content is an early or alternative research output and has not been peer-reviewed by Cambridge University Press at the time of posting.

Abstract

We refine the symmetric mean integral method for estimating the perimeter of an ellipse by restricting the integration limits to [0, π/4]. This approach allows the application of the squeeze theorem by leveraging the extremal behavior of the integrand, yielding explicit upper and lower bounds. The results provide a foundation for further research to derive improved perimeter estimates for ellipses.

Keywords

Elliptic integrals
Ellipse perimeter
Mean integral
Squeeze theorem
Symmetry

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Version History

Apr 29, 2025 Version 2

Version Notes

I have changed Theorem 2.1 to Lemma 2.1. This naming is more appropriate because it is useful for the main proof, namely Theorem 2.2. The wording of Lemma 2.1 and Theorem 2.2 has been corrected to make it more concise and easy to understand. In addition, I value the order of t that satisfies the maximum value for the function f(t).

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The content is available under CC BY NC ND 4.0 [opens in a new tab]

DOI

10.33774/coe-2025-5bb4j-v2
D O I: 10.33774/coe-2025-5bb4j-v2 [opens in a new tab]

Author’s competing interest statement

The author(s) have declared they have no conflict of interest with regard to this content

Ethics

The author(s) have declared ethics committee/IRB approval is not relevant to this content

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