Fast Isotopy Computation for T-Curves
2026-04-10 • Computational Geometry
Computational Geometry
AI summaryⓘ
The authors study T-curves, which are special shapes made from a grid-like pattern and signs, used to understand certain smooth curves drawn on a plane. They explain how to quickly figure out the exact shape type (called the real scheme) of these curves using a new fast algorithm. By using powerful graphics processors, this method can analyze billions of shapes every second, making large-scale studies possible. This algorithm helped the authors list all 121 types of degree seven curves created using T-curves.
T-curvedegreeunimodular triangulationlattice pointsViro's Patchworking Theoremambient isotopyreal schemeGPU accelerationreal plane projective algebraic curve
Authors
Zoe Geiselmann, Michael Joswig, Lars Kastner, Konrad Mundinger, Sebastian Pokutta, Christoph Spiegel, Marcel Wack, Max Zimmer
Abstract
A T-curve of degree $d$ is given by a regular unimodular triangulation of $d \cdot Δ_2$ together with a sign distribution on its lattice points. By Viro's Patchworking Theorem, this determines the ambient isotopy type (a.k.a. real scheme) of a smooth real plane projective algebraic curve of the same degree. We present a near-quadratic time algorithm for extracting that isotopy type from the triangulation and the signs. Through a GPU-accelerated implementation, this allows one to compute billions of real schemes per second, enabling exhaustive enumeration at scale. This algorithm was essential for our recent construction of all 121 real schemes of degree seven by T-curves.