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Perimeter of an Ellipse: Understanding Ramanujan's Approximation

Uday Shankar

Abstract

It is well known that there is no closed form analytic expression for the perimeter of an ellipse. In 1927, Srinivasa Ramanujan provides two approximations to the perimeter of an ellipse that are amazingly accurate. However, he does not provide an explanation of how he arrived at those expressions. In this paper, we will try to provide such an explanation that is likely how he derived those expressions. Using insights from our analysis, we improve on these approximations. To the best of our knowledge, ours is the first attempt to explain Ramanujan's ellipse perimeter formula and our approximation is uniformly better than his expression.

Perimeter of an Ellipse: Understanding Ramanujan's Approximation

Abstract

It is well known that there is no closed form analytic expression for the perimeter of an ellipse. In 1927, Srinivasa Ramanujan provides two approximations to the perimeter of an ellipse that are amazingly accurate. However, he does not provide an explanation of how he arrived at those expressions. In this paper, we will try to provide such an explanation that is likely how he derived those expressions. Using insights from our analysis, we improve on these approximations. To the best of our knowledge, ours is the first attempt to explain Ramanujan's ellipse perimeter formula and our approximation is uniformly better than his expression.
Paper Structure (22 sections, 1 theorem, 69 equations, 2 figures)

This paper contains 22 sections, 1 theorem, 69 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Exact Derivation
  3. Basic Identities
  4. Perimeter of Ellipse using the Arc Length Formula
  5. Derivation of Ramanujan's Approximations via Continued Fractions
  6. Continued Fraction Expansion of E(x)
  7. High-Level Algorithm Summary
  8. Derivation of $R_1(x)$
  9. Derivation of $R_2(x)$
  10. Improving on Ramanujan's Approximation - Attempt 1
  11. Motivation
  12. Required Correction
  13. Defining the Form of Perturbation
  14. Can we do Better? - Attempt 2
  15. Derivation of the Closed Form
  16. ...and 7 more sections

Key Result

Theorem 1

If $0 \leqslant u \leqslant 1$ then

Figures (2)

  • Figure 1: Direct comparison of Ramanujan's approximations $R_1(x)$ and $R_2(x)$, Cantrell's formula $C(x)$ and our approximations $A_1(x)$ and $A_2(x)$, against the target series $E(x)$ over the interval $[0,1)$. The remarkable accuracy of all functions makes them difficult to distinguish on this scale.
  • Figure 2: The relative error of the five approximations plotted on a logarithmic y-axis. The slopes near $x=0$ indicate the order of local accuracy. $A_2(x)$ demonstrates superior performance across the entire domain. Note that the y-axis is shown at log scale.

Theorems & Definitions (2)

  • Theorem 1
  • proof