What question did this study set out to answer?

This paper aims to derive the perimeter formula of an ellipse using projection techniques and to analyze its accuracy.

May 4, 2026Open Access

On the Perimeter of an Ellipse: A Projection-Based Derivation and Asymptotic Error Analysis

Key Points

This paper aims to derive the perimeter formula of an ellipse using projection techniques and to analyze its accuracy.
Combined projection and calculus to derive the ellipse perimeter.
Developed a summation approximation with an asymptotic error analysis.
Proved a convergence rate of $mathcal{O}(x^{-4})$.
The new method demonstrates a relative error largely independent of eccentricity when $e$ is not near 1.
The provided projection-based formula offers an alternative to traditional integration methods.

Abstract

This paper presents an alternative derivation of the perimeter formula of an ellipse by combining projection and calculus. It offers a different perspective compared to traditional integration methods. The purpose of this paper is to illustrate the geometrical properties of ellipses. In addition, it further develops the projection method to obtain a summation approximation based on projection, with an O (x^-4) convergence rate and a relative error mostly independent of the eccentricity when e is not near 1, which is proved rigorously via asymptotic error analysis.

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