{
  "abstract": "A smooth plane algebraic curve is a compact Riemann surface and a real surface of codimension 2 in CP^2. Most methods that mesh it work in an affine chart or over a branched covering of CP^1, and must therefore treat points at infinity, charts or branch cuts separately. We triangulate the curve directly inside CP^2, with no distinguished chart. A 15-vertex triangulation of CP^2, due to Gaifullin, is refined adaptively by Maubach bisection along Fubini-Study geodesics, and the curve is followed from cell to cell by continuation. In each cell, the curve meets the triangular 2-faces in isolated points, which are joined into polygons. Realizing every face as a geodesic cone, rather than as a flat triangle in an affine chart, makes the faces shared by neighboring cells coincide exactly, so the pieces glue into a closed surface, whose topology is checked a posteriori. On our test curves at depth 14, fewer than 0.14% of the cells that meet the curve fail, against 1.8-11.6% with chart-flat faces; the area agrees with Wirtinger's theorem to within 0.15%; and, unlike a box in an affine chart at equal resolution, the mesh covers the whole curve.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "name": "Vinícius Mello"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e9fygbcxtqd4n.bib",
  "cited_by": 0,
  "doi": null,
  "fulltext_url": null,
  "id": "e9fygbcxtqd4n",
  "issns": [],
  "landing_url": "https://hal.science/hal-05791116v1",
  "language": "en",
  "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": [
    [
      "hal",
      "0.info"
    ],
    [
      "hal",
      "0.math"
    ],
    [
      "hal",
      "1.info.info-cg"
    ],
    [
      "hal",
      "1.math.math-ag"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://hal.science/hal-05791116/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e9fygbcxtqd4n.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:53Z",
      "native_id": "hal-05791116",
      "source": "hal",
      "url": "https://hal.science/hal-05791116v1"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "hal:1.math.math-ag",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Triangulating Plane Algebraic Curves Intrinsically in CP^2",
  "title_text": "Triangulating Plane Algebraic Curves Intrinsically in CP^2",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:53Z",
  "url": "https://research.draconitic.com/paper/e9fygbcxtqd4n",
  "venue": "HAL",
  "versions": []
}