{
  "abstract": "The closest-point problem for a circular helix reduces to Kepler’s equation u+esinu=M with geometric eccentricity e=aρ/b2 unrestricted above. Building on this known reduction and earlier half-turn localization, we derive an exact stationary-point count and a theorem selecting the global minimizer. For e≤1, the squared distance is strictly convex and has a unique strict minimum, including the degenerate boundary case. For e>1, the generic count is N=2(M+c)/2π−2(M−c)/2π−3, where c=arccos(−1/e)+e2−1. Separate indicators give the distinct and multiplicity counts on every fold, including simultaneous double roots. The count grows like 2e/π; five distinct stationary points become possible precisely when e>4.6033…. We express the classical normal-plane envelope, called the focal surface here, as explicit bifurcation sheets and determine the cut locus and its termination at the evolute helix. The principal-branch theorem identifies the nearest-turn root as the unique global minimizer with exactly two minimizers when M≡π(mod2π) and e>1. For fixed e>1, its absolute root condition number is bounded by 1/(e−1). Fold separation and cusp displacement scale as square and cube roots of the data offset; the corresponding condition numbers diverge with exponents −1/2 and −2/3. A bisection–Newton solver, an equal-pitch two-helix reduction, and reproducible numerical checks complement the analysis.",
  "abstract_shown": true,
  "affiliations": [
    "School of Mathematics, Northeastern University, Shenyang 110819, China"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Peilin Luo"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bmh0x3yw75hqa.bib",
  "cited_by": 0,
  "doi": "10.3390/math14203674",
  "fulltext_url": null,
  "id": "bmh0x3yw75hqa",
  "issns": [
    "2227-7390"
  ],
  "landing_url": "https://www.mdpi.com/2227-7390/14/20/3674",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "openalex",
      "2608|Mathematics and Applications"
    ]
  ],
  "openalex": "W7221568017",
  "pdf_url": null,
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-10",
  "publisher": "MDPI AG",
  "ris": "https://research.draconitic.com/paper/bmh0x3yw75hqa.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:29Z",
      "native_id": "10.3390/math14203674",
      "source": "crossref",
      "url": "https://www.mdpi.com/2227-7390/14/20/3674"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2608|Mathematics and Applications",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Closest Points on a Circular Helix and Kepler’s Equation: Exact Counting, the Focal Surface, and the Principal-Branch Minimizer",
  "title_text": "Closest Points on a Circular Helix and Kepler’s Equation: Exact Counting, the Focal Surface, and the Principal-Branch Minimizer",
  "type": "peer-reviewed",
  "updated_at": "2026-10-30T12:18:29Z",
  "url": "https://research.draconitic.com/paper/bmh0x3yw75hqa",
  "venue": "Mathematics",
  "versions": []
}