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An Exponential Correction to Ramanujan's Second Formula for Ellipse Perimeter Computation
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DOI：10.3390/appliedmath6040056.png)
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Abstract
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The exact perimeter of an ellipse involves the complete elliptic integral of the second kind, which lacks a closed-form expression in elementary functions. As a result, analytical approximations have been proposed for applications requiring fast and accurate evaluation of elliptical geometries. In this study, we present a new ultra-accurate and compact closed-form approximation for the ellipse perimeter based on an exponential correction applied to Ramanujan's second formula. The proposed expression preserves simplicity-using only three exponential functions and six constants-while achieving a maximum relative error of approximately 0.57 ppm observed over the tested grids covering the full eccentricity range. This represents a significant accuracy improvement over classical and modern approximations while maintaining a single-line analytical form with low computational cost. Due to its robustness, quasi-exact behavior at both circular and highly eccentric limits, and its suitability for numerical algorithms and embedded implementations, the proposed approximation is particularly useful in engineering computations involving elliptical boundaries.
Keywords:
closed-form ellipse perimeter approximation
Ramanujan II approximation
elliptical orbit of comets
approximation to ellipse circumference
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0.7
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111
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