Otakhon U. Kenjaev

Arc length of an elliptical helix via Ramanujan's second perimeter approximation, with explicit pitch thresholds

We observe that 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 acc...