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On the Perimeter of an Ellipse - mklabgr - 07-28-2026
On the Perimeter of an Ellipse
Summary
In the paper "On the Perimeter of an Ellipse" (published in The Mathematica Journal), author Paul Abbott examines exact expressions and efficient numerical approximations for calculating an ellipse's perimeter, demonstrating how Mathematica's computational capabilities simplify these derivations. Abbott shows that historical exact formulations—including those by Maclaurin, Euler, and Gauss–Kummer—are interconnected via quadratic hypergeometric transformations, yielding symmetric identities in terms of elliptic integrals and Legendre functions.
He then explores a variety of approximation techniques, ranging from polynomial and Chebyshev interpolation to rational minimax approximants, placing special emphasis on "extreme perfect" approximants that remain exact for both a circle and a fully flattened ellipse. Ultimately, the work illustrates how combining symbolic transformations, special function identities, numerical optimization, and visualization in Mathematica naturally leads to highly accurate, computationally efficient formulas for ellipse perimeters.
ARTICLE [PDF]