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.