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

Abstract

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 accurate. We give 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. The result gives integral-free wire and path lengths for elliptical coil springs, elliptical helical antennas and helical machining paths. Version 2 adds an exact formula for coils with stepped pitch and part turns, and closed-form estimates for progressive-pitch and tapered elliptical coils. Use of generative AI: Claude Opus 5.5 (Anthropic, claude-opus-5-5) was used to suggest the question, to carry out the symbolic and high-precision numerical checks (SymPy, mpmath) under the author's direction, and to help prepare the text. The author selected the problem, checked all results and takes full responsibility for the content.