Triangulating Plane Algebraic Curves Intrinsically in CP^2
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.
Related
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BibTeX
@misc{mello2026triangulating,
title = {{Triangulating Plane Algebraic Curves Intrinsically in CP^2}},
author = {Vinícius Mello},
howpublished = {HAL},
year = {2026},
month = oct,
url = {https://hal.science/hal-05791116v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Triangulating Plane Algebraic Curves Intrinsically in CP^2
AU - Vinícius Mello
PY - 2026
DA - 2026/10/10
PB - HAL
UR - https://hal.science/hal-05791116v1
AB - 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.
ER -