Ellipse perimeter approximation: New formula with absolute relative error less than one ppm.

Abstract

In this article, the author presents two modifications to his original formula for Ellipse Perimeter Approximation, published two years ago. The objective is to reduce the Absolute Relative Error further. After the first modification, the Absolute Relative Error becomes less than one millimetre per kilometre, (that is to the order of 10-7) for ellipses with eccentricities ranging from zero to 0.92. Though for the remaining ellipses, with eccentricities between 0.92 and 1, this modification increases the Absolute Relative Error up to the order of 10-4, it is reduced and contained within the order of 10-6 by an error approximation formula. The second modification limits the Absolute Relative Error to less than 5 centimetres per kilometre across all eccentricities.