Elliptic integrals · Geometry
Arc length of an elliptical helix via Ramanujan's second perimeter approximation, with explicit pitch thresholds
\[L_{\text{turn}} = P\!\left(\sqrt{a^2+c^2},\ \sqrt{b^2+c^2}\right)\]
One turn of the elliptical helix is exactly an ellipse perimeter — so Ramanujan's formula gives it in closed form.
- \(c \ge 0.197254\,a \Rightarrow \text{rel. error} < 10^{-6}\)
- \(c \ge 0.534032\,a \Rightarrow \text{rel. error} < 10^{-9}\)
Abstract
One turn of the elliptical helix r(t) = (a cos t, b sin t, ct) has exactly the length of the ellipse with semi-axes sqrt(a^2+c^2) and sqrt(b^2+c^2). Consequently Ramanujan's second approximation for the perimeter of an ellipse gives a closed-form turn length, and the pitch makes the approximation more accurate. Explicit thresholds: if c >= 0.197254 a the relative error is below 10^-6 for every b in [0,a]; if c >= 0.534032 a it is below 10^-9.
Cite
Otakhon U. Kenjaev (2026). Arc length of an elliptical helix via Ramanujan's second perimeter approximation, with explicit pitch thresholds. Zenodo. https://doi.org/10.5281/zenodo.22969890
@misc{kenjaev2026ellipticalhelix,
author = {Kenjaev, Otakhon U.},
title = {Arc length of an elliptical helix via Ramanujan's second perimeter approximation, with explicit pitch thresholds},
year = {2026},
publisher = {Zenodo},
doi = {10.5281/zenodo.22969890}
}