Error Reduction in Koshy’s Formula for Ellipse Perimeter: Correction Factor and Improved Approximation

K. Idicula Koshy¹

Section:Research Paper, Product Type: Journal-Paper
Vol.7 , Issue.1 , pp.1-8, Feb-2020

Online published on Feb 28, 2020

Copyright © K. Idicula Koshy . This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
 

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Abstract :
In an article published in IJESIRD, the author has introduced a new empirical formula for ellipse perimeter. He observed that the definite integral for the ellipse perimeter satisfies the first order partial differential equation known as Lagrange’s equation and, therefore, the perimeter must be a function of two appropriate independent solutions of the partial differential equation. Though the actual formula could not be derived, he developed an empirical formula. The maximum absolute relative error in the perimeter as per the formula is less than 6.0*10-5 across all aspect ratios b/a from 0 to 1. In the present article, the author improves his formula, reducing the relative error further to less than (5.93)*10-6 by means of a Correction Factor. When all aspect ratios are considered together, the Improved Perimeter Formula, compared with other formulae of its category, yields lowest ‘maximum relative error and error fluctuation’. Surprisingly, for ellipses with aspect ratios less than 0.15, the relative error due to the Improved Perimeter Formula is less than that due to S. Ramanujan’s Formula II. Moreover, the formula is simple enough for computation using a standard scientific calculator.

Key-Words / Index Term :
Ellipse, Major/Minor Radii, Aspect Ratio, Simpson’s Rule, Relative error, Correction Factor

References :
[1] K. I. Koshy, “Ellipse Perimeter: A new Approximate Formula by a New Approach”, International Journal of Engineering Science Invention Research and Development (IJESIRD), Vol. VI, Issue 1, 2019.
[2] Erwin Kreyszig, “Advanced Engineering Mathematics”, 8th Ed., Wiley, Ch.17, 2010.
[3] S. Ramanujan, “Modular Equations and Approximations to π”,Quarterly Journal. of Math.(Oxford) Vol. 45, pp 350-372, 1914.
[4] Mark B Villarino, “Accuracy of Ramanujan’s Formula for Perimeter of an Ellipse”, Journal of inequalities in Pure and Applied Math. Vol. 1, Issue1, pp 1-10, 2006.