arXiv subjects

Manuel Rendón-Marín

Publications and source records attributed to Manuel Rendón-Marín.

2 recordsLinked to original sources

arXiv2026

Approximating the Ellipse Perimeter by "Bending the Line": The ELIP Closed-Form Family

The perimeter of an ellipse is expressed exactly through the complete elliptic integral of the second kind, motivating accurate closed-form approximations. We introduce ELIP (Ellipse Linear Interpolated Perimeter), a family built by correcting the straight line joining the two exact normalized endpoint values. Two branches are developed: logarithmic-power corrections with optional deformation and exponential modulators, and rational-logarithmic corrections incorporating the exact asymptotic behavior of highly elongated ellipses. The framework distinguishes coefficients fixed analytically by Maclaurin matching from parameters optimized numerically by minimax fitting. Compact rationalized formulas achieve maximum relative errors of approximately 6.84, 3.57, 0.904, and 1.596 ppm. An ELIP R5,2 + 3 log formula with three fitted degrees of freedom has a certified maximum relative error below 0.001623 ppm. For ELIP R5,3 + 3 log, all rational-core coefficients are determined by exact Maclaurin matching. A one-parameter multiplicative correction, 1 - A*h^12*(1-h)^(7/4), reduces the certified global error bound to below 0.000452 ppm. This formula matches the exact perimeter series through h^8, while the correction starts at order h^12. The results provide a systematic approach combining exact endpoint geometry, local series constraints, elongated-limit logarithmic asymptotics, and low-dimensional numerical optimization.

math.GM↗

arXiv2025

Is simplicity still possible for a more accurate approximation to the perimeter of the ellipse? or, Using the exponential function to further improve the second Ramanujan's approximation

The perimeter of an ellipse has no exact closed-form expression in terms of elementary functions, and numerous approximations have been proposed since the eighteenth century. Classical formulas by Fagnano, Euler, and Ramanujan, as well as modern refinements such as Cantrell and Koshy methods, aim to reduce the approximation error while maintaining computational simplicity. In this paper, we introduce a new closed-form expression that enhances Ramanujan second formula by dividing it by 1 minus a binomial of two exponential terms resulting in a very stable approximation in a range of b/a between 1 and 1/10000, or even up to a smaller ratio. The resulting approximation remains compact, requiring only four constants, and achieving a remarkable tradeoff between simplicity and accuracy. Across the full eccentricity range of b/a in [0.0001,1], our method attains a maximum relative error of approximately 0.57 ppm with respect to the exact perimeter computed via elliptic integral. Our formula is quasi-exact at the extremes, for the circle b/a=1 and for the degenerate flat ellipse b/a=0. Compared with Cantrell approximation, the proposed method reduces the maximum relative error by a factor of 25 while preserving a short and elegant expression. This makes it one of the simplest yet most accurate closed-form and single-line approximations to the ellipse perimeter currently available in the literature.

math.NA↗