{
  "abstract": "Starting from a geometric observation about the perimeter of an ellipse, a normalized function c(x) = 2ln(4/x)·AGM(1,x) is introduced that expresses the arithmetic-geometric mean as a smooth monotone curve interpolating between two transcendental constants: π as x→0 and 4ln2 at x=1. The boundary values and slopes of this curve are read directly from the asymptotic behavior of the complete elliptic integral K at its two limiting regimes. These four conditions uniquely determine a parameter-free power-law approximation with exponent α = (2−2ln2)/(π−4ln2) ≈ 1.6631, composed entirely of π and ln2 with no free parameters. The approximation achieves a maximum error of −0.1636% near x ≈ 0.3466, improving to −0.0001% near the degenerate limit x→0. The error is one-signed throughout the domain; the conjecture that the approximation is a strict lower bound is supported by analytic characterization at both endpoints and interval arithmetic verification to 154 decimal digits at 9,999 interior points. Part of a series: Ellipse perimeter · Nonlinear pendulum · Quartic anharmonic oscillator · Asymmetric cubic oscillator · Relativistic harmonic oscillator",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "name": "William Lennon"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dwx52afp6bha2.bib",
  "cited_by": 6,
  "doi": "10.5281/zenodo.20464351",
  "fulltext_url": null,
  "id": "dwx52afp6bha2",
  "issns": [],
  "landing_url": "https://doi.org/10.5281/zenodo.20464351",
  "language": "en",
  "licence": "CC-BY-SA",
  "licence_label": "CC BY-SA",
  "licence_source": "journal",
  "licence_url": "https://creativecommons.org/licenses/by-sa/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "openalex",
      "2604|Mathematical functions and polynomials"
    ]
  ],
  "openalex": "W7162857735",
  "pdf_url": null,
  "pmcid": null,
  "pmid": null,
  "published": "2026-05-30",
  "publisher": "European Organization for Nuclear Research",
  "ris": "https://research.draconitic.com/paper/dwx52afp6bha2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:17Z",
      "native_id": "W7162857735",
      "source": "openalex",
      "url": "https://doi.org/10.5281/zenodo.20464351"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2604|Mathematical functions and polynomials",
      "subfield": "general"
    }
  ],
  "title": "A Geometric Route to a Parameter-Free Approximation of the Arithmetic-Geometric Mean",
  "title_text": "A Geometric Route to a Parameter-Free Approximation of the Arithmetic-Geometric Mean",
  "type": "preprint",
  "updated_at": "2026-11-06T13:20:22Z",
  "url": "https://research.draconitic.com/paper/dwx52afp6bha2",
  "venue": "Zenodo (CERN European Organization for Nuclear Research)",
  "versions": []
}