Wire needed for a coil wound on an oval or elliptical former. Exact numerical length next to the closed-form estimate from Ramanujan's ellipse formula, with the pitch folded into the semi-axes.
| Turns | Pitch, mm |
|---|
Lengths are measured along the wire centreline, so the semi-axes are a = (W − d)/2 and b = (H − d)/2.
For the helix (a cos t, b sin t, ct) with pitch p = 2πc, the pitch is added in quadrature to both semi-axes. One turn has exactly the length of the plane ellipse with
Ramanujan's second approximation gives P in closed form; the pitch makes the ellipse rounder, so the estimate gets more accurate as the pitch grows.
Stepped pitch: each section of nᵢ turns at pitch pᵢ contributes nᵢ·P(Aᵢ, Bᵢ); a part turn is an incomplete elliptic integral A·E(φ, m), m = 1 − B²/A².
Progressive pitch: Simpson's rule over each turn on the one-turn formula, with the pitch at the start, middle and end of the turn.
Conical coil: one-turn formula at mid-turn with A² = a² + β² + c², B² = b² + α² + c², where α, β are the radial taper per radian. For a round cone the length is exact:
Cite as: Kenjaev, O. U. (2026). Arc length of an elliptical helix via Ramanujan's second perimeter approximation, with explicit pitch thresholds (Version 2). Preprint, Zenodo. doi:10.5281/zenodo.22984129 · doi.org/10.5281/zenodo.22984129 · all versions: 10.5281/zenodo.22969890 · ORCID 0009-0009-3566-9285 · Telegram @sheki · channel @nizomliy